73 


IN  MEMORIAM 
FLOR1AN  CAJORI 


STANDARD  ARITHMETIC 


EMBRACING  A  COMPLETE  COURSE  FOR  SCHOOLS  AND 
ACADEMIES 


BY 
WILLIAM   J.   MILNE,   PH.D.,  LL.D. 

i\ 
PRESIDENT  OF  NEW  YORK  STATE  NORMAL  COLLEGE,  ALBANY,  N.Y. 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN   BOOK   COMPANY 


MILNE'S    MATHEMATICS 

ARITHMETIC 

PROGRESSIVE   THREE   BOOK   SERIES 

First  Book $0.35 

Second  Book 40 

Third  Book 45 

PROGRESSIVE  TWO   BOOK   SERIES 

First  Book        .  35 

Complete  Book 65 

STANDARD  TWO   BOOK   SERIES 

Elements  of  Arithmetic     .......         .30 

Standard  Arithmetic 65 

STANDARD   THREE   BOOK   SERIES 

Primary  Arithmetic ........         .25 

Intermediate  Arithmetic 30 

Standard  Arithmetic 65 

MENTAL  ARITHMETIC        . 35 

ALGEBRA    AND  GEOMETRY 

GRAMMAR   SCHOOL  ALGEBRA 50 

ELEMENTS   OF  ALGEBRA 60 

HIGH   SCHOOL  ALGEBRA i.oo 

ACADEMIC  ALGEBRA          .......       1.25 

ADVANCED  ALGEBRA 1.50 

STANDARD   ALGEBRA i.oo 

PLANE  AND   SOLID   GEOMETRY 1.25 

PLANE  GEOMETRY.     Separate 75 

SOLID   GEOMETRY.     Separate 75 

Copyright,  1892  and  1895,  by  AMERICAN  BOOK  COMPANY. 
Stand.  Ar. 

E-P      131 


PEEFACE. 


IN  the  preparation  of  this  work  the  author  has  aimed  to 
secure  two  results  ;  namely,  skill  in  numerical  computations, 
and  a  proper  understanding  of  the  reasons  for  the  steps  in 
the  explanation  of  processes  and  the  solution  of  problems. 
Skill  in  computing  may  be  acquired  without  any  intelligent 
apprehension  of  arithmetical  science,  and  a  profound  insight 
into  the  truths  and  principles  of  arithmetic  may  be  attained 
without  much  facility  in  using  numbers.  Very  many  people 
will  prefer  to  have  the  student  trained  to  be  rapid  and 
accurate  in  computations,  and  they  will  esteem  a  rapid 
accountant  more  competent  in  mathematics  than  the  learned 
astronomers  of  our  time;  while  others  will  prefer  that 
training  which  cultivates  the  reasoning  powers,  even  at 
the  expense  of  practical  expertness  in  the  use  of  numbers. 
The  author  has  endeavored  to  secure  both  these  ends  by 
embodying  in  the  book  a  large  number  of  examples  upon 
which  the  pupil  may  be  trained  to  accuracy  and  rapidity, 
while  at  the  same  time  he  has  not  failed  to  incorporate  in 
it  a  large  number  of  problems  that  are  designed  to  train 
the  analytical  powers  and  to  develop  the  reasoning  faculties. 

In  practical  business  life,  the  processes  learned  in  schools 
are  often  of  very  little  value,  because  they  are  not  the  nat- 
ural processes  of  the  business  man.  Students  who  learn 
to  work  examples  in  a  mechanical  way  find  themselves 
unable  to  solve  with  certainty  very  simple  problems,  after 
they  have  left  school  a  few  weeks,  because  they  have  been 
taught  a  school  method  rather  than  a  natural  method. 
The  author  has,  therefore,  adopted  business  methods  of 
computation  wherever  they  could  be  wisely  substituted  for 

3 


4  PREFACE. 

the  processes  of  the  schools:  he  has  preceded  them  with 
exercises  which  lead  the  student  directly  and  easily  to  a 
clear  apprehension  of  the  steps  in  the  solution  and  the 
necessity  for  them ;  and  he  has  accompanied  the  solutions 
with  explanations  which  enable  the  pupil  to  comprehend 
all  that  he  needs  to  know  about  the  operations.  By  these 
means  the  student  is  led  to  employ  business  methods  of 
solution  because  they  are  generally  natural  methods,  and 
to  understand  and  explain  every  step  in  the  process.  A 
student  who  has  been  trained  in  this  manner  will  never 
forget  a  process  or  a  rule,  because  he  can  devise  the  process 
and  frame  the  rule  at  will. 

The  work  is  of  sufficiently  comprehensive  scope  to  meet 
the  demands  of  even  the  most  advanced  schools.  The 
unusually  practical  character  of  the  problems  will  be  dis- 
cerned by  a  very  cursory  examination;  oral  and  written 
exercises  are  given  in  connection  with  each  subject,  and 
frequent  and  thorough  reviews  serve  to  test  the  pupil's 
proficiency,  to  fix  the  principles  of  the  science  in  his  mind, 
and  to  train  and  develop  his  power  of  reasoning. 

The  method  exemplified  in  presenting  the  various  subjects 
is  in  accord  with  what  is  deemed  best  in  modern  methods 
of  teaching;  the  order  and  arrangement  of  the  subjects, 
though  they  are  in  some  respects  a  departure  from  that 
usually  given,  will  hasten  the  pupil's  progress  by  removing 
to  the  latter  part  of  the  book  subjects  too  difficult  for  the 
average  pupil  when  he  reaches  them,  and  of  little  practical 
value  to  any  student;  the  explanations  are  thought  to  be 
conspicuously  lucid,  the  steps  logical,  the  definitions,  prin- 
ciples, and  rules  brief  and  accurate. 

The  author  desires  to  express  his  indebtedness  to  many 
educators  of  prominence  for  valuable  suggestions  regarding 
the  scope  of  the  work  and  its  educational  character.  The 
cordiality  with  which  his  former  works  have  been  received 
gives  him  the  hope  that  this  book  also  may  meet  with 
general  favor. 

WILLIAM  J,  MILNE, 


CONTENTS. 


PAGE 

NOTATION  AND  NUMERATION 9 

The  Arabic  System 9 

United  States  Money 18 

The  Roman  System 19 

ADDITION 21 

SUBTRACTION 38 

MULTIPLICATION 52 

DIVISION 67 

Relation  of  Dividend,  Divisor,  and  Quotient 84 

Analysis  and  Review 85 

FACTORS 91 

Tests  of  Divisibility 93 

Factoring 94 

Cancellation 95 

FRACTIONS 99 

Reduction 102 

Addition 110 

Subtraction 114 

Multiplication 117 

Division 122 

Fractional  Forms •. 127 

Fractional  Relation  of  Numbers 128 

Review  Exercises 131 

DECIMAL  FRACTIONS 144 

Notation  and  Numeration 144 

Reductioi. . ,                                           148 


6  CONTENTS. 

DECIMAL  FRACTIONS.  —  Continued. 

Addition 151 

Subtraction 153 

Multiplication 155 

Division 158 

Short  Processes • 161 

Accounts  and  Bills 164 

Review  Exercises 166 

DENOMINATE  NUMBERS 170 

Reduction 171 

Addition 189 

Subtraction 191 

Multiplication 193 

Division 194 

Review  Exercises 196 

LONGITUDE  AND  TIME 201 

PRACTICAL,  MEASUREMENTS 205 

GENERAL  REVIEW  EXERCISES 220 

PERCENTAGE 231 

Profit  and  Loss 246 

Commission 251 

Commercial  Discount 253 

Taxes 255 

Duties  or  Customs 257 

Insurance 259 

INTEREST 262 

Annual  Interest 269 

Compound  Interest 270 

Promissory  Notes 273 

Partial  Payments 276 

Problems  in  Simple  Interest 279 

True  Discount -  282 

Bank  Discount 284 


CONTENTS. 


INTEREST.  —  Continued. 

Stocks  and  Bonds  ......................................  289 

Review  Exercises  .......................................  296 

EXCHANGE  .................................................  301 

Domestic  Exchange  .....................................  303 

Foreign  Exchange  ......................................  306 

PARTNERSHIP  ...............................................  308 

RATIO  .....................................................  312 

PROPORTION  ................................................  314 

Simple  Proportion  ......................................  317 

Compound  Proportion  ...................................  320 

Partitive  Proportion  .....................................  323 

INVOLUTION  ................................................  324 

EVOLUTION  ....................................    ............  327 

Evolution  by  Factoring  ..................................  327 

Square  Root  ............................................  328 

Applications  of  Square  Root  .............................  332 

Similar  Surfaces  ........................................  334 

Cube  Root  .............................................  335 

Applications  of  Cube  Root  ...............................  341 

Similar  Volumes  ........................................  341 

GENERAL  REVIEW  EXERCISES  ...............................  343 

AVERAGE  OF  PAYMENTS  .....................................  368 

AVERAGE  OF  ACCOUNTS  .....................................  372 

SAVINGS  BANK  ACCOUNTS  ...................................  374 

PROGRESSIONS  ...............................    ..............  376 

Arithmetical  Progression  ................................  377 

Geometrical  Progression  .................................  379 

PROBLEMS  IN  COMPOUND  INTEREST  ...........................  381 

ANNUITIES  ........................................  .........  383 

DIVISORS  AND  MULTIPLES  ...................................  386 

Common  Divisors  .......................................  386 

Common  Multiples  ......................................  389 


8  CONTENTS. 

DIVISORS  AND  MULTIPLES.  —  Continued.  PAGE 

Greatest  Common  Divisor  of  Fractions 392 

Least  Common  Multiple  of  Fractions 393 

CIRCULATING  DECIMALS 394 

SCALES  OP  NOTATION 397 

PROOFS 400 

Fundamental  Processes 400 

DIVISION  BY  FACTORS 402 

MEASUREMENT  OF  SOLIDS 403 

Convex  Surface  of  Solids 405 

Volume  of  Solids 408 

METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES 411 

Metric  Tables 412 

Reduction 414 

TABLES  OF  DENOMINATE  NUMBERS 418 

Measures  of  Extension 418 

Measures  of  Capacity 421 

Measures  of  Weight 422 

Measures  of  Time 425 

Circular  or  Angular  Measure 426 

Measures  of  Value 427 

Counting 430 

Stationers'  Table 430 

INTEREST  AND  PARTIAL"  PAYMENTS 431 

Vermont  Rules 431 

New  Hampshire  Rule 434 

Connecticut  Rule 436 

TAXES « 437 


STANDARD     ARITHMETIC. 


NOTATION   AND   NUMERATION. 


1.  A  single  thing  is  called  a  Unit. 

2.  A  unit  or  a  collection  of  units  is  a  Number. 
A  number  answers  the  question  "  how  many  ?  " 

A  number  may  be  expressed  by  words  or  other  characters, 
viz.  figures  and  letters. 

3.  The  method  of  expressing  numbers  by  figures  or  let- 
ters is  called  Notation. 

The  method  of  expressing  numbers  by  figures  is  called  the  Arabic 
Notation,  from  the  Arabs  who  first  introduced  it  into  Europe. 

The  method  of  expressing  numbers  by  letters  is  called  the  Roman 
Notation,  because  it  was  used  by  the  ancient  Romans. 

4.  The  method  of  reading  numbers  expressed  by  figures 
or  letters  is  called  Numeration. 

THE    ARABIC    SYSTEM. 

5.  In  counting  a  large  number  of  objects,  it  is  natural  to 
arrange  them  in  equal  groups.     When  the  number  of  the 
first  groups  becomes  large  they  may  be  gathered  into  larger 
groups,  and  these  again  into  larger  groups,  and  so  on.     By 
general  agreement  the  system  of  grouping  by  tens,  called 
the  decimal  system,  has  been  adopted. 

9 


10  NOTATION  AND  NUMERATION. 

6.  The  Arabic  system  of  notation,  which  is  a  decimal 
system,  employs  ten  figures  to  express  numbers,  viz. : 

0123456789 
Naught    One     Two    Three    Four     Five       Six     Seven    Eight    Nine 

Naught  is  also  called  zero  and  cipher. 
By  combining  these  figures  in  accordance  with  certain 
principles,  any  number  can  be  expressed. 

7.  PRINCIPLE.  —  When  figures  are  written  side  by  side, 
the  one  at  the  right  expresses  units,  the  next  tens,  and  the  next 
hundreds. 

EXERCISES. 

8.  Tell  what  each  figure  in  the  following  expresses : 

43  37  57  186  453  304 

36  86  48  371  416  215 

81  29  73  218  830  507 

32  51  63  134  591  290 

9.  Figures  in  units'  place  express  units  of  the  first  order; 
those  in  tens'  place,  units  of  the  second  order ;  those  in  hun- 
dreds' place,  units  of  the  third  order ;  etc. 

10.  The  units  of  the  second  order,  or  tens,  are  named  ten, 
twenty,  thirty,  forty,  fifty,  sixty,  seventy,  eighty,  ninety. 

The  suffix  ty  means  ten.     Thus  forty  means  four  tens. 

11.  The  numbers  between  1  ten  and  2  tens  are  named 
eleven,  twelve,   thirteen,  fourteen,  fifteen,   sixteen,   seventeen, 
eighteen,  nineteen. 

Thirteen  means  three  and  ten  ;  fourteen,  four  and  ten,  etc. 

12.  The  other  numbers  between  20  and  100  are  read  with- 
out the  word  and  between  the  tens  and  the  units. 

Thus,  35  is  read  thirty-five,  not  thirty  and  five. 


THE  ARABIC   SYSTEM. 


11 


EXERCISES. 
13.   Read  the  following : 


18 

46 

54 

49 

35 

98 

39 

15 

14 

38 

93 

17 

85 

27 

34 

72 

23 

69 

24 

65 

40 

22 

20 

50 

66 

79 

30 

77 

25 

43 

61 

83 

16 

28 

99 

33 

12 

21 

45 

31 

44 

80 

Express  by  figures  the  following  : 
Seventy-five.  Forty-five.  Eighty. 

Eighty-six.  Fifty-eight.  Eighteen. 

Thirty-nine.  Eighty-one.  Seventy-eight. 

Ninety-eight.  Twenty-four.  Thirty-three. 

Seventy-seven.          Fifty-nine.  Ninety-nine. 

Twenty-one.  Ninety-one.  Sixty-four. 

Three  units  of  the  second  order,  five  of  the  first  order. 
Five  units  of  the  second  order,  seven  of  the  first  order. 
Seven  units  of  the  first  order,  nine  of  the  second  order. 
Write  all  numbers  below  twenty. 
Write  all  numbers  between  twenty  and  forty. 
Write  all  numbers  between  fifty  and  seventy. 

14.  In  reading  numbers  expressed  by  three  figures,  the 
tens  are  read  after  the  hundreds,  and  the  units  after  the 
tens,  without  the  word  and. 

Thus,  346  is  read  three  hundred  forty-six. 


EXERCISES. 
15.   Read  the  following : 

442  815  844  600  765 

378  763  419  408  811 

426  341  906  391  501 


12  NOTATION  AND  NUMERATION. 

Kead  the  following : 

927  594  836  903  700 

538  318  379  830  555 

421  423  221  712  279 

376  873  718  127  380 

673  465  178  104  308 

894  347  187  199  300 

Express  by  figures  the  following  : 

Seven  hundred  forty-five.          One  hundred  three. 

Eight  hundred  eighty-four.      Two  hundred  ninety-five. 

Six  hundred  forty-eight.  Seven  hundred. 

Nine  hundred  fifty-nine.  Nine  hundred  ninety-nine. 

Five  hundred  eighty-one.          Four  hundred  five. 

Six  hundred  four.  Two  hundred  eighteen. 

One  hundred  eighty-one.  Six  hundred  nine. 

Eight  hundred  fifty.  Eight  hundred  eighty. 

Two  hundred  fifteen.  Nine  hundred. 

Three  hundreds,  six  tens,  eight  units. 

Eight  hundreds,  four  units. 

Seven  hundreds,  eight  tens. 

Five  units  of  the  third  order,  two  of  the  second,  three  of 
the  first. 

Two  units  of  the  third  order,  two  of  the  second,  two  of 
the  first. 

Four  units  of  the  third  order,  four  of  the  second,  four  of 
the  first. 

16.  From  the  previous  examples  the  following  general 
principle  is  deduced : 

PRINCIPLE.  —  The  representative  value  of  a  figure  is  in- 
creased ten-fold  by  each  removal  one  place  to  the  left,  and 
decreased  ten-fold  by  each  removal  one  place  to  the  right. 


THE   ARABIC   SYSTEM.  13 

17.  In  writing  and  reading  numbers,  the  figures  are  sepa- 
rated  into   groups   of  three   figures   each,    called  periods. 
These  periods  contain  the  hundreds,  tens,  and  units  of  each 
denomination. 

18.  The  following  table  illustrates  the  system  of  notation : 


PERIODS.  6th..  5th..  4th..          3d.  2d.          1st. 

NAMES  o  rf  g 

OF  s  "    1  i  I 

PERIODS.  B  §  t 


ORDERS.          .   g          .  g          .  g          .  g          .   g  g 

i  j-  I    i  P  i    z  £  I    *  £  I    x  £  i    i  i-  i 

25,   673,  210,  040,  385,   861 


The  number  is  read  twenty-five  quadrillion,  six  hundred 
seventy-three  trillion,  two  hundred  ten  billion,  forty  million, 
three  hundred  eighty-Jive  thousand,  eight  hundred  sixty-one. 

1.  Each  period,  except  the  one  at  the  left,  must  contain  three  figures. 

2.  The  periods  are  separated  from  each  other  by  commas. 

3.  In  reading  numbers  the  name  of  units'  period  is  omitted. 

4.  The  periods  above  quadrillions  in  their  order  are  quintillions, 
sextillions,  septillions,  octillions,  nonillions,  decillions,  etc. 

RULE  FOR  NUMERATION.  —  Beginning  at  the  right,  sepa- 
rate the  numbers  into  periods  of  three  figures  each. 

Beginning  at  the  left,  read  each  period  as  if  it  stood  alone, 
adding  its  name. 

EXERCISES. 
19.   Copy,  point  off  into  periods,  and  read : 

1.  3825,          42865,          346812,          18573912. 

2.  1713,          31793,          386045,          43056784. 


14 


NOTATION  AND  NUMERATION. 


3. 

4651, 

84275, 

713204, 

4. 

3042, 

80163, 

504216, 

5. 

2104, 

25068, 

800437, 

6. 

3640, 

40016, 

504036, 

7. 

5812, 

32004, 

240040, 

8. 

7346, 

41500, 

213045, 

9. 

7604, 

68314, 

508023, 

10. 

6150, 

65036, 

700016, 

11. 

2738, 

42050, 

910006, 

12. 

3807, 

71685, 

380460, 

13. 

3943, 

91804, 

402040, 

14. 

5008, 

87005, 

180247, 

15. 

6900, 

34506, 

707365, 

16. 

7165, 

71346, 

205006, 

36700431. 

45320406. 

781003042. 

451320645. 

800400300. 

729009001. 

738040000. 

1245679316. 

4506780259. 

37009854629. 

870053126945. 

650030012503. 

19876005012036. 

47020100316042. 


EXERCISES. 
20.   Write  in  figures : 

1.    Twenty-nine  billion,  ninety-five  thousand,  forty-five. 

EXPLANATION.  —  We  first  write  the  numbers  of  the  highest  denom- 
ination,  following  them  with   a    comma.    After  that  we  write  the 
Cumbers  of  the  next  lower  denomination,  or  mil- 

lions?  but  gince  tbere  are  none^  we  m  the  period 

with  ciphers,  and  write  a  comma  after  them.  This  is  followed  by 
095  in  the  next  lower  period,  and  045  in  the  last,  the  figure  0  being 
placed  before  the  significant  figures  to  make  the  periods  complete. 

EULE  FOR  NOTATION.  —  Begin  at  the  left  and  write  the 
hundreds,  tens,  and  units  of  each  period  in  their  proper  order, 
putting  ciphers  in  all  vacant  places  and  periods. 

While  writing,  separate  each  period  by  a  comma  from  the 
one  that  follows  it. 


THE  ARABIC   SYSTEM.  15 

Write  in  figures  : 

2.  Twenty-five  thousand,  eight  hundred  fourteen. 

3.  Thirty-nine  thousand,  seven  hundred  twenty-four. 

4.  Ninety-four  thousand,  six  hundred  fifty-five. 

5.  Twenty-nine    thousand,    five    hundred    eighty-five. 
TCight  thousand,  six  hundred  five. 

6.  Forty-six  thousand,  eight   hundred  twenty.     Forty 
thousand,  eight  hundred  nineteen. 

7.  Eighty-four  thousand,  nine  hundred  four.      Fifteen 
thousand,  nine.     Eight  thousand,  four  hundred. 

8.  Nineteen  thousand,  nine  hundred  nine.     Four  thou- 
-sand,  eight.     Eighteen  thousand,  seven  hundred. 

9.  Twenty-four  thousand,  two  hundred  eight.     Twenty 
thousand.     Thirty  thousand,  seven  hundred. 

10.  Fifteen  thousand,  four  hundred  seven.     Four  thou- 
.sand,  seven.     Twenty  thousand,  two  hundred. 

11.  Thirty-seven  thousand,  nine  hundred.     Thirty  thou- 
sand, thirty.     One  hundred  thousand,  eight  hundred  five. 

12.  Fifty-five  thousand,  five  hundred  five.    Five  thousand, 
five.     Thirty  thousand,  nine  hundred  nine. 

13.  Forty-eight    thousand,   fifty-five.      Forty  thousand, 
eight.     Seventeen  thousand,  seven  hundred  three. 

14.  Sixty-six    thousand,   eighty.      Fifty-five    thousand, 
nine.     Fifty  thousand,  three  hundred  eighty-five. 

15.  Eighty-five  thousand,  eight  hundred  eight.     Eighty- 
eight  thousand.     Eight  thousand,  eight  hundred  eight. 

18.  Thirty  thousand,  three  hundred  thirty-five.  Twenty 
thousand,  nine.  Ninety  thousand,  two  hundred  eight. 

17.  Two  hundred  eighteen  thousand,  five  hundred  sixty- 
.seven.  Eighty  thousand,  seven  hundred  twenty. 


16  NOTATION  AND  NUMERATION. 

18.  Four   hundred   thirty-three   thousand,    six  hundred 
fifty-five.     Fifty-five  thousand,  eight  hundred  nine. 

19.  Five  hundred  forty-three  thousand,  eight  hundred 
seventy-six.     Three  hundred  ten  thousand. 

20.  Nine  hundred  ninety  thousand,  two  hundred  nine. 

21.  Five  hundred  fifty  thousand,  eight  hundred  four. 

22.  Six  hundred  sixty  thousand,  two  hundred  fifteen. 

23.  Seven    hundred    thousand,    eighty.     Two    hundred 
thousand,  five.     Six  hundred  forty  thousand. 

24.  Eight  hundred  twenty-five  thousand,  seven  hundred 
eight.     Fifty  thousand,  five  hundred  five. 

25.  Two  hundred  forty  thousand,  six  hundred  eighty-five. 

26.  Nine  hundred  ninety-seven  thousand,  four  hundred. 

27.  Six  hundred  thousand,  eight  hundred  eighty-nine. 

28.  Nine    hundred    thousand,    nine    hundred.      Seven 
hundred  thousand.     Ninety-five  thousand,  five. 

29.  Four  hundred  sixteen  thousand,  two  hundred  twenty. 

30.  Three  hundred  eighty  thousand,  five  hundred  fifty. 

31.  Four  hundred  fifty  thousand,  three  hundred  eighty. 

32.  Three  hundred  eighty-six  thousand,  forty-seven. 

33.  One  hundred  twenty-eight   million,  three   hundred 
twenty-eight  thousand,  six  hundred  fifty-seven. 

34.  Five    hundred   twenty-seven    million,    six    hundred 
eighteen  thousand,  two  hundred  sixty-four. 

35.  Two  hundred  forty-three  million,  four  hundred  sixty- 
seven  thousand,  eight  hundred  sixty-nine. 

36.  Forty-five  million,  two  hundred  thirty-four  thousand, 
six  hundred  ninety-four. 

37.  Two  hundred  eighteen  million,  eighty-four  thousand, 
eight  hundred  fifteen. 


THE   ARABIC   SYSTEM.  17 

38.  39  million,  46  thousand,  90;  24  million,  180  thou- 
sand, 340 ;  49  million,  18  thousand,  20. 

39.  212  million,  206  thousand,  8;   91  million,  87  thou- 
sand, 65  ;  37  million,  8  thousand,  206. 

40.  526  million,  300  thousand,  80 ;  418  million,  40  thou- 
sand, 210;  408  million,  48  thousand,  48. 

41.  312  million,  115  thousand,  116;  40  million,  80  thou- 
sand, 80 ;  250  million,  105  thousand,  37. 

42.  206  million,  6  thousand,  6 ;  505  million,  40  thousand, 
40 ;  315  million,  75  thousand,  75. 

43.  400  million,  40  thousand,  40;   60  million,  60  thou- 
sand, 60 ;  360  million,  265. 

44.  50  million ;  200  million,  200 ;  300  million,  3. 

45.  215  billion,  618  million,  415  thousand,  816. 

46.  236  billion,  212  million,  836  thousand,  309. 

47.  454  billion,  369  million,  800  thousand,  80. 

48.  500  billion,  41  million,  41  thousand,  45. 

49.  613  billion,  40  million,  40  thousand,  40. 

50.  Forty  billion,  eighty  million,  nine  thousand,  eighty. 

51.  Twenty  billion,  two  hundred  million,  ten  thousand, 
ten. 

52.  One  hundred  billion,  one  million,  one  thousand,  one. 

53.  Four  trillion,  three  hundred  six  billion,  four  hun- 
dred eight  million,  two  hundred  twenty  thousand,  forty. 

54.  Twenty-eight  trillion,  two  hundred  billion,  forty-six 
million,  eight  hundred  forty  thousand,  two  hundred  fifty. 

55.  Seventy-five  trillion,  two  hundred  billion,  two  hun- 
dred million,  two  hundred  thousand,  two  hundred. 

56.  Eight  hundred  trillion,  eight  billion,  eight  million, 
eight  hundred  thousand,  eighty. 

STAND.    AR. 2 


18  NOTATION  AND  NUMERATION. 

NOTATION   AND   NUMERATION   OP   UNITED 
STATES   MONEY. 

21.  The  currency  of  the  United  States  has  a  decimal 
system  of  notation,  dollars  being  written  as  whole  numbers 
and  cents  as  decimal  parts  of  a  dollar. 

22.  The  dollar  sign  is  $.     It  is  written  before  the  num- 
ber. 

Thus,  $  25  is  read  twenty-five  dollars. 

23.  In  notation  of  United  States  currency  a  period,  called 
the  decimal  point,  is  placed  before  the  cents. 

The  dollars  are  written  at  the  left  of  the  decimal  point. 
The  first  two  places  at  the  right  of  the  decimal  point 
express  cents,  and  the  third,  tenths  of  a  cent  or  mills. 

Thus,  $10.485  is  read  ten  dollars,  forty-eight  cents,  five  mills. 

24.  When  the  number  of  cents  is  less  than  ten,  a  cipher 
must  be  written   in   the   first  place  at  the   right   of   the 
decimal  point. 

Thus,  Five  dollars  five  cents  is  written  $5.05. 

EXERCISES. 

25.  Read  the  following : 

1.  $52.28,  $212.84,'  $200.855,  $7004.275. 

2.  $13.71,  $358.16,  $196.327,  $3648.032. 

3.  $16.04,  $427.24,  $518.043,  $36845.92. 

4.  $15.90,  $600.85,  $508.405,  $29043.06. 

5.  $43.09,  $693.05,  $700.049,  $3104.066. 

6.  $37.86,  $410.30,  $326.416,  $21040.30. 


THE  ROMAN  SYSTEM.  19 

26.  Write  the  following : 

1.  Fifteen    dollars,    twelve    cents.      Eighteen    dollars, 
eight  cents.     Twenty-four  dollars,  eight  cents. 

2.  Thirty-four  dollars,  thirty  cents.     Fifty-five  dollars, 
twenty  cents.     Nineteen  dollars,  thirty-eight  cents. 

3.  Forty  dollars,  four  cents.     Ninety-nine  dollars,  nine 
cents.     Sixty-four  dollars,  eleven  cents. 

4.  Four  hundred  dollars,  eight  cents.     Seven  hundred 
dollars.     Seventeen  dollars,  eight  cents,  eight  mills. 

5.  Two  hundred  thirty-eight  dollars,  twenty  cents,  five 
mills.     Ninety -three  dollars,  forty  cents,  eight  mills. 

6.  Three  hundred  ninety-one  dollars,  forty-eight  cents, 
three  mills.  Sixty-seven  dollars, sixty-seven  cents, three  mills. 

7.  Four  thousand  three  hundred  twenty  dollars,  eight 
cents.     Fifty-nine  dollars,  twenty  cents,  seven  mills. 

8.  One  thousand  two  hundred  forty-nine  dollars,  nine 
cents,  five  mills.    Forty-seven  dollars,  ninety  cents,  five  mills. 

9.  Eighty-four  thousand  three   hundred   dollars,  nine 
cents.     Thirty  dollars,  nine  cents,  nine  mills. 

10.    Fifty-five  thousand  eight  hundred  sixteen   dollars, 
five  cents.     One  thousand  dollars,  ten  cents,  five  mills. 

THE   ROMAN   SYSTEM. 

27.  This  system  uses  seven  capital  letters   to   express 
numbers,  viz.: 

Letters,    I,      V,      X,      L,        C,         D,          M. 
Values,     ±,      5,      10,     50,     10O,     500,     10OO, 


20 


NOTATION  AND  NUMERATION. 


23.   The  following  principles  are  followed  in  combining 
the  letters : 

PRINCIPLES.  —  1.   Repeating  a  letter  repeats  its  value. 
Thus,  I  represents  one  ;  II,  two ;  III,  three  ;  X,  ten ;  XX,  twenty. 

2.  When  a  letter  is  placed  before  another  of  greater  value, 
its  value  is  to  be  taken  from  that  of  the  greater. 

Thus,  IV  represents  four ;  IX,  nine  ;  XIX,  nineteen ;  XL,  forty. 

3.  When  a  letter  is  placed  after  another  of  greater  value, 
their  values  are  to  be  united. 

Thus,  VII  represents  seven ;  XV,  fifteen  ;  LXXX,  eighty. 

4.  A  bar  placed  over  a  letter  increases  its  value  a  thousand- 
fold. 

Thus,    V  represents  five  thousand;    L,  fifty  thousand;   M,  one 
million. 

The  following  table  illustrates  the  method  of  combination. 


I  . 

.  1 

X  . 

.  10 

XXIV 

.  .  24 

C  . 

.     100 

II  . 

.  2 

XI  . 

.  11 

XXIX 

.  .  29 

CC  . 

.     200 

Ill  . 

.  3 

XIV  . 

.  14 

XXX 

.  .  30 

cccc  . 

.     400 

IV  . 

.  4 

XV  . 

.  15 

XL 

.  .  40 

CD  . 

.     400 

V  . 

.  5 

XVI  . 

.  16 

L 

.  .  50 

D  . 

.     500 

VI  . 

.  6 

XIX  . 

.  19 

LX 

.  .  60 

DCCC  . 

.     800 

VII  . 

.  7 

XX  . 

.  20 

LXX 

.  .  70 

M  . 

.  1000 

IX  . 

.  9 

XXI  . 

.  21 

XC 

.  .  90 

MMM  . 

.  3000 

EXERCISES. 
29.   Bead  the  following  numbers : 

XVII;  XXV;  LXXX;  XIX;  XXIX;  XLV;  CXV; 
XCV;  LXXIX;  CXIX;  XCIX ;  _XLIV;  CCCIV; 
CCXLIV;  CCCCXC;  DCCLXXXIX;  XDCCCLXXII. 

Express  the  following  numbers  by  the  Roman  notation : 
13,  24,  71,  68,  132,  514,  244,  555,  617,  1040,  7216,  2899. 


ADDITION. 


30.  1 .   How  many  apples  are  3  apples  and  2  apples  ? 

2.  How  many  books  are  3  books  and  4  books  ? 

3.  How  many  leaves  are  2  leaves  and  3  leaves  ? 

4.  How  many  oranges  are  4  oranges  and  2  oranges  ? 

5.  What  have  you  been  doing  with  the  numbers  given 
above  ? 

6.  Why  can  you  not  tell  how  many  5  cents  and  4  rabbits 
are? 

7.  What  kind  of  numbers  only  can  be  united  ? 

31.  The  process  of  rinding  a  number  which  is  equal  to 
two  or  more  given  numbers  is  called  Addition. 

32.  The  result  obtained  by  adding  is  called  the  Sum,  or 
Amount. 

33.  The  numbers  added  are  called  Addends. 

34.  The  Sign  of  Addition  is  an  upright  cross  +.     It  is 
called  plus,  and  is  placed  between  the  numbers  to  be  added. 

Thus,  3  +  7  is  read  3  plus  7,  and  it  means  that  3  and  7  are  to  be 
added. 

35.  The  Sign  of  Equality  is  two  equal  short  horizontal 
lines :  =.     It  is  read  equals  or  is  equal  to. 

Thus,  3  +  7  =  10  is  read  3  plus  7  equals  10. 

21 


22  ADDITION. 

36.  Any  expression  of  equality  is  called  an  Equation. 
Thus,  3  +  7  =  10  and  5  +  4  =  9  are  equations. 

37.  Numbers    that   have  the  same  unit  are  called  Like 
Numbers. 

Thus,  $7  and  $5  are  like  numbers;  so  also  are  15  pounds  and  8 
pounds. 

38.  PRINCIPLES.  —  1.    Only  like  numbers  can  be  added. 
2.    The  sum  and  the  addends  must  be  like  numbers. 

DRILL  EXERCISES. 

39.  The  student  should  practice  adding  the  following 
numbers  daily  until  he  can  tell  the  sums  at  a  glance. 

The  list  contains  all  the  combinations  of  two  numbers 
from  1  to  9. 

3          4          2          6          8.2          6          2          5 
7          5          6          3          19          1          2          7 


623314786 
473829157 


312769867 
518465857 


654324341 
858154234 


584937997 
192198969 


ORAL   EXERCISES.  23 

ORAL  EXERCISES. 

40.   1.   Harry  paid  5  cents  for  a  pencil  and  10  cents  for 
a  writing  book.    How  much  did  he  pay  for  both  ?    5+10=  ? 

2.  Mary  learned  8  new  words  on  Monday  and  9  on  Tues- 
day.    How  many  did  she  learn  on  both  days  ?     8  +  9  =  ? 

3.  James  earned  $3  in  May,  $4  in  June,  and  $6  in 
July.      How  much  did  he  earn   in   the   three   months  ? 
3+4+6=? 

4.  I  gave  5  apples  to  my  sister,  6  to  my  brother,  and 
then   had    7    for    myself.     How    many  had    I    at    first? 
5+6+7=? 

5.  A  teacher  gave  for  a  lesson  on  Monday  6  problems,  on 
Tuesday  7,  and  on  Wednesday  8.     How  many  did  she  give 
in  the  three  days  ?     6  +  7  +  8=? 

6.  Mary  put  into  her  bank  5  cents  at  one  time,  8  cents 
at  another,  and  10  at  another.     How  much  did  she  put  in 
altogether  ? 

7.  A  lad  saw  three*  flocks  of  wild  geese.     In  the  first 
there  were  9,  in  the  second  7,  and  in  the  third  10.     How 
many  were  there  in  all  ? 

8.  Sarah's  locket  cost  $  8,  the  chain  $  6,  and  her  ring 
$  10.     How  much  did  they  all  cost  ?    8  +  6  +  10=? 

9.  A  gentleman  owned  9  gray  horses,  7  black  ones,  and 
10  bay  ones.   How  many  horses  did  he  own  ?   9  +  7  +  10  =  ? 

10.  A  bookseller  one  day  sold  8  first  readers,  4  second 
readers,  and  5  third  readers.    How  many  readers  did  he  sell  ? 

11.  A  boy  rode  5  miles  and  back  on  his  bicycle,  and  then 
walked  3  miles.     How  far  did  he  travel  ?     5  +  5  +  3=? 

12.  A  farmer  planted  three  fields  with  corn,  the  first 


24  ADDITION. 

containing  9  acres,  the  second  5  acres,  and  the  third  7  acres. 
How  many  acres  of  corn  did  he  plant  ?     9  +  5  +  7=? 

13.  Samuel  caught  8  rats  with  one  trap,  5  with  another, 
and  6  with  another.  How  many  rats  did  he  catch? 
8+5+6=? 

41.  Count  or  add  by  2's  from  0  to  30,  thus :  0,  2,  4,  6,  etc. 

Add  by  2's  from  1  to  31,  thus ;  1,  3,  5,  7,  etc. 

Add  by  3's  from  0  to    36.  From  1  to  43. 

Add  by  3's  from  2  to    47.  From  3  to  45. 

Add  by  4's  from  0  to    48.  From  1  to  61. 

Add  by  4's  from  2  to    50.  From  3  to  63. 

Add  by  5's  from  0  to    60.  From  1  to  61. 

Add  by  5's  from  2  to    52.  From  3  to  63. 

Add  by  5's  from  4  to    54.  From  7  to  82. 

Add  by  6's  from  0  to    60.  From  1  to  55. 

Add  by  6's  from  2  to    56.  From  3  to  63. 

Add  by  6's  from  4  to    58.  From  5  to  59. 

Add  by  7's  from  0  to    70.  From  1  to  78. 

Add  by  7's  from  2  to    79.-  From  3  to  87. 

Add  by  7's  from  4  to    88.  From  5  to  96. 

Add  by  7's  from  6  to    97.  From  9  to  100. 

Add  by  8's  from  0  to    80.  From  1  to  81. 

Add  by  8's  from  2  to    90.  From  3  to  91. 

Add  by  8's  from  4  to    92.  From  5  to  101. 

Add  by  8's  from  6  to  102.  From  7  to  103. 

Add  by  9's  from  0  to    90.  From  1  to  100. 

Add  by  9's  from  2  to  101.  From  3  to  111. 

Add  by  9's  from  4  to  103.  From  5  to  113. 

Add  by  9's  from  6  to  114.  From  7  to  106. 

Add  by  9's  from  8  to  116.  From  9  to  126. 
Add  6  to  35,  45,  55,  65,  75,  85,  95. 
Add  7  to  38,  48,  58,  68,  78,  88,  98. 


WRITTEN  EXERCISES.  25 

Add  8  to  36,  46,  56,  66,  76,  86,  96. 

Add  5  to  33,  43,  53,  63,  73,  83,  93. 

Add  9  to  39,  49,  59,  69,  79,  89,  99. 

Add  4  to  37,  47,  57,  67,  77,  87,  97. 

WRITTEN    EXERCISES. 

42.   Copy  and  add  from  the  bottom  upwards,  and  then 
from  the  top  downwards,  each  of  the  following : 

In  adding,  name  results  only.     Thus  in  example  1  add  as  follows  : 
2,  6,  15,  22,  25  instead  of  2  and  4  are  6,  6  and  9  are  15,  etc. 

1.       2.       3.       4.       5.       6.       7.       8.       9.      10.     11.     12. 


3 

5 

4 

6 

8 

6 

5 

6 

4 

6 

5 

7 

7 

6 

5 

2 

2 

6 

8 

8 

3 

4 

8 

9 

9 

2 

3 

7 

9 

3 

4 

3 

3 

8 

7 

3 

4 

7 

8 

5 

5 

8 

7 

9 

8 

2 

6 

1 

2 

9 

1 

3 

7 

6 

2 

4 

7 

5 

4 

5 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

24. 

5 

6 

3 

5 

8 

5 

8 

8 

3 

5 

4 

8 

6 

8 

9 

4 

3 

7 

3 

8 

9 

4 

3 

8 

8 

4 

7 

6 

9' 

6 

2 

7 

2 

3 

8 

9 

7 

9 

6 

8 

2 

9 

9 

7 

1 

7 

5 

9 

9 

3 

8 

7 

1 

8 

5 

6 

8 

6 

7 

4 

2 

5 

4 

3 

9 

4 

1 

6 

2 

8 

9 

4 

3 

8 

9 

4 

5 

7 

6 

6 

9 

5 

6 

3 

25. 

26. 

27. 

28. 

29. 

30. 

31. 

32. 

33. 

34. 

35. 

36. 

7 

4 

3 

4 

9 

1 

5 

9 

8 

5 

4 

4 

4 

5 

2 

3 

2 

2 

5 

8 

1 

6 

4 

9 

9 

3 

3 

4 

3 

3 

5 

4 

4 

8 

5 

1 

3 

4 

5 

3 

7 

4 

6 

5 

6 

4 

3 

8 

8 

6 

4 

7 

1 

5 

6 

3 

5 

3 

8 

2 

€ 

8 

7 

6 

5 

6 

6 

1 

7 

9 

7 

9 

5 

3 

6 

1 

4 

7 

7 

2 

3 

6 

6 

3 

26  ADDITION. 

Find  the  sum  of  the  following : 

(a)  (6)  (c)  (d)  («)  (/)  (<?)  (A)  (<)  (j)  (ft)  (Z) 

37.  7,  8,  3,  6,  4,  5,  1,  2,  9,  8,,  4,  6. 

38.  6,  5,  4,  6,  3,  8,  7,  5,  4,  7,  3,  5. 

39.  3,  9,  6,  5,  4,  1,  9,  3,  2,  3,  5,  4. 

40.  8,  1,  6,  3,  4,  3,  6,  7,  8,  2,  8,  8. 

41.  5,  4,  9,  3,  8,  9,  6,  1,  7,  1,  7,  4. 

42.  8,  3,  1,  9,  4,  2,  2,  6,  4,  4,  1,  3. 

43.  4,  1,  3,  2,  6,  8,  8,  3,  5,  6,  6,  7. 

44.  2,  7,  9,  1,  8,  5,  7,  3,  6,  3,  4,  9. 

45.  9,  1,  6,  8,  4,  9,  3,  2,  1,  8,  3,  6. 

46.  8,  3,  4,  2,  6,  8,  5,  6,  9,  7,  7,  4. 

47.  1,  3,  8,  4,  3,  2,  5,  4,  8,  2,  8,  7. 

48.  5,  9,  8,  4,  2,  7,  6,  8,  3,  6,  2,  5. 

49.  3,  7,  8,  4,  5,  3,  6,  9,  5,  5,  7,  4. 

50.  9,  5,  9,  8,  7,  6,  5,  4,  2,  5,  3,  2. 

(TO)  (n)  (o)  00  (g)  (r)  (s)  («)  (M)  (i)  (w)  (x) 

51.  3,  5,  4,  3,  3,  8,  4,  6,  5,  7,  3,  5. 

52.  4,  2,  2,  9,  6,  5,  2,  9,  7,  8,  5,  8. 

53.  2,  9,  5,  5,  4,  3,  8,  4,  2,  9,  6,  3. 

54.  8,  8,  8,  7,  5,  7,  2,  6,  3,  9,  1,  7. 

55.  7,  3,  2,  5,  1,  8,  9,  2,  4,  5,  7,  6. 

56.  6,  4,  6,  8,  2,  8,  3,  6,  5,  7,  6,  9. 

57.  9,  5,  3,  1,  5,  2,  4,  5,  8,  2,  7,  5. 

58.  5,  3,  9,  7,  7,  8,  3,  4,  2,  6,  5,  8. 

59.  2,  8,  4,  4,  1,  8,  5,  4,  3,  1,  4,  4. 

60.  3,  1,  8,  2,  7,  9,  2,  6,  5,  4,  2,  2. 

61.  1,  7,  2,  3,  5,  4,  3,  9,  8,  7,  8,  3. 

62.  8,  9,  9,  8,  2,  6,  3,  1,  8,  4,  7,  9. 

63.  7,  2,  3,  4,  5,  2,  6,  8,  5,  4,  1,  1. 

64.  6,  5,  5,  2,  5,  4,  3,  9,  2,  7,  6,  5. 


ORAL  EXERCISES.  27 

Add  the  columns  as  follows  : 


65. 

(a). 

71. 

(9)- 

77. 

(m). 

83. 

w- 

66. 

(6). 

72. 

(ft). 

78. 

(«)• 

84. 

(0. 

67. 

(c). 

73. 

(0. 

79. 

(0). 

85. 

(«). 

68. 

(<*). 

74. 

CO- 

80. 

(*). 

86. 

(V). 

69. 

(e). 

75. 

Car). 

81. 

(<# 

87. 

(w). 

70. 

(/)• 

76. 

(0- 

82. 

(r). 

88. 

(a). 

ORAL    EXERCISES. 

43.  1.  A  certain  school  had  30  boys  and  40  girls  in 
attendance.  How  many  pupils  were  there  in  the  school  ? 

2.  A  boy  paid  25  cents  for  a  reading  book  and  55  cents 
for  an  arithmetic.     How  much  did  he  pay  for  both  ? 

3.  January  has  31  days  and  February  usually  28.     How 
many  days  are  there  from  January  1  to  March  1  ? 

SUGGESTION.  —  Add  by  uniting  the  tens  and  units  separately. 
Thus,  31  +  20  =  51 ;  61  +  8  =  59.  Or  30  +  20  =  50,  8  +  1  =  9,  and 
60  +  9  =  59. 

4.  A  boy  who  is  now  12  years  of  age  is  35  years  younger 
than  his  father.     How  old  is  his  father  ? 

5.  A  railway  train  ran  35  miles  the  first  hour  and  33  miles 
the  second  hour.     How  far  did  it  run  in  the  two  hours  ? 

6.  I  bought  a  pound  of  tea  for  45  cents  and  some  raisins 
for  32  cents.     How  much  did  I  pay  for  both  ? 

7.  A  farmer  sold  three  loads  of  potatoes,  the  first  con- 
taining 25  bushels,  the  second  25  bushels,  and  the  third  20 
bushels.    How  many  bushels  did  he  sell  ?    25  +  25  +  20  =  ? 

8.  A  lady  paid  $25  for  a  cloak,  $15  for  a  dress,  and 
$  20  for  blankets.     How  much  did  she  pay  for  all  ? 

9.  I  traveled  35  miles  by  railroad,  25  miles  on  a  steam- 
boat, and  15  miles  by  stage.    How  many  miles  did  I  travel  ? 

SUGGESTION.  —  Add  thus:  35  +  20  =  55;  55  +  5  =  60;  60  +  15  =  75. 


28  ADDITION. 

10.  Henry  owned  22  Leghorn   chickens,  30  Plymouth 
Kocks,  and  12  bantams.     How  many  chickens  did  he  own  ? 

11.  A  cotton  merchant  bought  30  bales  on  Monday,  25 
on  Tuesday,  and  33  on  Wednesday.     How  many  did  he  buy 
in  all  ? 

12.  A  lad  sold  30  morning  papers,  40  evening  papers, 
and  21  illustrated  papers.     How  many  papers  did  he  sell  ? 

13.  The  distance  from  Brent  to  Afton  is  15  miles  ;  from 
Afton  to  Sandley,  30  miles ;  from  Sandley  to  Darmel,  40 
miles.     How  far  is  it  from  Brent  to  Darmel  ? 

14.  In  an  intermediate  school  I  counted  25  pupils  who 
read  in  the  third  reader,  35  who  read  in  the  fourth  reader, 
and  36  who  read  in  the  fifth   reader.     If  these  were   all 
the  pupils,  how  many  were  there  in  the  school  ? 

15.  Three  girls  made  paper  dolls  for  a  fair;   the  first 
made  15,  the  second  made  22,  the  third  made  20.     How 
many  did  they  all  make  ? 

16.  When  I  arose  the  temperature  was  47  degrees ;  an 
hour  later  it  was  13  degrees  higher ;  and  at  noon  it  was  17 
degrees  higher  still.     What  was  the  temperature  at  noon  ? 

17.  A  grocer  had  three  barrels  of  molasses,  the  first  con- 
taining 51  gallons,  the  second  44,  and  the  third  50.     How 
many  gallons  did  they  all  contain  ? 

18.  A  merchant  sold  three  webs  of  cloth,  the  first  con-; 
taming  37  yards,  the  second  43  yards,  and  the  third  30. 
yards.     How  many  yards  did  he  sell  ? 

19.  A  student  misspelled  45  words  the  first  term  of  the 
year,  30  the  second  term,  and  25  the  third.     How  many 
were  misspelled  by  him  during  the  year  ? 

20.  The  Empire  State  express  train  ran  52  miles  in  one 
hour,  61  miles  the  next  hour,  and  53  miles  the  next  hour. 
How  far  did  it  travel  in  the  three  hours  ? 


WRITTEN   EXERCISES.  29 


WRITTEN    EXERCISES. 

44.   1.   What  is  the  sum  of  $  394,  $  476,  and  $  549  ? 

EXPLANATION.  —  For  convenience  in  adding,  the  numbers  are 
arranged  so  that  units  of  the  same  order  stand  in  the  same  ver- 
tical column. 

Each  column  is  then  added  separately,  beginning  with 
the  right  hand  column,  or  units.    Thus,  9  +  6  -f  4  =  19, 

SR  ^QJ.     the  SUm  °f  the  units-     19  units  are  equal  to  1  ten  and  9 

units.    The  9  is  therefore  written  under  the  units'  column, 

476     and  the  1  is  reserved  to  add  with  the  tens. 

549          1  reserved  +  4  +  7  +9  =  21,  the  sum  of  the  tens.    21 

—     tens  are  equal  to  2  hundreds  and  1  ten.     The  1  is  therefore 

$  1419     written  under  the  tens'  column,  and  the  2  is  reserved  to  add 

with  the  hundreds. 

2  reserved  +  5  +  4  +  3  =  14,  the  number  of  hundreds.     14  hun- 
dreds are  equal  to  1  thousand  and  4  hundreds,  and  are  written  in 
thousands'  and  hundreds'  places  in  the  sum. 
Hence  the  sum  is  $  1419. 

1.  In  adding,  only  the  results  should  be  named.     Thus,  instead  of 
saying  9  and  6  are  15  and  4  are  19,  add  in  the  following  manner :  9, 
15,  19. 

2.  When  the  sum  of  any  column  is  10,  20,  or  some  exact  number 
of  tens,  a  cipher  is  written  under  the  column  added,  and  the  1,  2,  or  3, 
etc.,  is  reserved  to  add  with  the  next  column. 

RULE.  —  Arrange  the  numbers  so  that  the  units  of  the  same 
order  stand  in  the  same  column. 

Beginning  at  the  right,  add  each  column  separately  and  write 
the  sum,  if  it  is  less  than  ten,  under  the  column  added. 

If  the  sum  of  any  column  is  ten  or  more,  write  the  unit 
Jigure  only  under  that  column,  and  add  the  tens  with  the  next 
column. 

Write  the  entire  sum  of  the  last  column. 

PROOF.  —  Add  each  column  in  the  reverse  order.  If  the 
results  agree,  the  work  is  probably  correct. 


30  ADDITION. 

Copy,  add,  and  prove : 


2. 

3. 

4.       5. 

6. 

7. 

416 

328 

265     796 

834 

$4.37 

325 

419 

783     843 

912 

9.28 

843 

327 

248     685 

897 

7.36 

794 

818 

415     792 

685 

8.94 

8. 

9. 

10.     11. 

12. 

13. 

487 

384 

$3.85   $4.37 

$2.38 

$6.66 

95 

296 

4.16     2.16 

4.51 

5.84 

385 

818 

5.86    5.05 

2.18 

.73 

486 

47 

.79     7.26 

3.09 

.81 

793 

93 

3.24     .83 

.92 

.92 

43 

786 

5.87     .87 

.85 

1.69 

861 

695 

4.83     .91 

4.65 

9.99 

459 

888 

5.12    4.55 

5.93 

4.00 

14. 

15. 

16. 

17. 

18. 

3297 

8568 

3972 

4184 

$24.36 

2935 

3864 

4136 

3879 

12.48 

4683 

3979 

5864 

4987 

63.23 

5987 

6846 

4839 

5998 

47.84 

3869 

5976 

8645 

7986 

84.38 

6984 

7143 

9256 

8438 

34.24 

19. 

20. 

21. 

22. 

23. 

4837 

4896 

6868 

$38.16 

$58.79 

5192 

1543 

4739 

24.34 

21.83 

7185 

9834 

8314 

73.58 

56.91 

3925 

8567 

7934 

48.15 

63.82 

3987 

8919 

9256 

41.93 

84.68 

7981 

9124 

8567 

98.34 

91.94 

8426 

9395 

8394 

90.09 

87.58 

5155 

8304 

8040 

39.90 

16.85 

WRITTEN  EXERCISES. 


31 


24. 

25. 

26. 

27. 

28. 

3857 

4168 

$53.87 

$32.54 

$15.92 

5879 

3925 

4.83 

16.87 

4.86 

389 

4095 

15.00 

8.34 

.76 

1089 

5683 

3.64 

.58 

43.41 

83 

7925 

.98 

13.24 

6.88 

1396 

84 

42.79 

24.48 

15.62 

3859 

8 

.09 

72.16 

9.24 

518 

385 

4.79 

8.43 

.89 

29. 

30. 

31. 

32. 

33. 

$3.965 

$23.14 

$9.875 

$11.18 

$45.83 

4.18 

51.07 

3.186 

24.16 

2.155 

5.465 

18.05 

7.24 

5.865 

.875 

9.235 

31.94 

8.39 

3.19 

4.18 

8.015 

41.06 

9.157 

24.54 

24.35 

7.07 

38.94 

8.386 

9.573 

21.375 

6.155 

48.358 

4.58 

18.46 

18.469 

34.  Add  3985,  4168,  3975,  4189,  2853,  9168. 

35.  Add  3854,  2198,  3864,  593,  786,  4153,  841. 

36.  Add  5841,  2976,  9183,  24,  863,  719,  818. 

37.  Add  3985,  268,  39403,  218,  36,  3965,  824. 

38.  Add  453,  2795,  68,  3920,  7,  8596,  873,  91. 

39.  Add  2168,  421,  3973,  263,  114,  3846,  200,  392. 

40.  Add  385,  2964,  372,  916,  84,  849,  9327,  56,  294. 

41.  Add  936,  1728,  494,  1672,  18,  5148,  614,  792,  371. 

42.  Add  7342,  8165,  294,  38,  297,  1986,  24,  386,  48,  24. 

43.  Add  3759,  2836,  384,  2795,  73,  295,  3865,  362,  51. 

44.  Add  415,  38,  2968,  504,  37,  3958,  26,  394,  283,  64. 

45.  Add  5168,  3796,  584,  317,  29,  38,  47,  6847,  365,  84. 


32 


ADDITION. 


46.  Add  342,  81,  8,  9164,  395,  19,  8,  4865,  96,  26,  813. 

47.  Add  49,  3812,  468,  9834,  275,  86,  391,  19,  35,  696. 

48.  Add  581,  27,  3986,  423,  4917,  2846,  9,  18,  48,  297. 

49.  Add  5396,  84,  4193,  875,  431,  97,  86,  8436,  938,  19. 

50.  Add  35,  2965,  47,  3894,  483,  769,  85,  94,  46,  324,  8. 


Copy,  add,  and  prove 


51. 

298576 
382453 
321294 
514685 
687349 
593426 
583724 
417938 
526867 
493256 
684325 
796853 
467397 
685486 
567834 


52. 

2795843 
3865487 
4975836 
9843725 
7659814 
8537685 
9132586 
8149375 
4396854 
9198374 
8168937 
9857694 
8123586 
3712584 
4865392 


53. 

38467589 
21438758 
47368975 
86123874 
98375692 
85399137 
92468546 
76856789 
39845926 
94398765 
85673423 
41798659 
98376954 
39257697 
82512763 


54. 

241693798 
473185429 
391583768 
427936857 
819348673 
473925165 
274639827 
315987352 
675431298 
897316984 
931258469 
687316984 
931258469 
716874912 
843978517 


55.  Add  236  thousand,   8   hundred  85;   118  thousand, 
9  hundred  27 ;  46  thousand,  8  hundred  95 ;  246  thousand, 

7  hundred  17. 

56.  Add   8   million,   324  thousand,   7  hundred   96;   15 
million,   289    thousand,   4   hundred  85;    91    million,   825 
thousand,  4  hundred  12 ;  15  million,  116  thousand,  8  hun- 
dred 96. 

57.  Add  28  million,  16  thousand,  875;  46  million,  324 
thousand,  536;  39  million,  413  thousand,  39;   24  million, 

8  thousand,  18 ;  8  million,  8  thousand,  8. 


WRITTEN  EXERCISES.  33 

58.  Add  5  million,  816  thousand,  4;  291  million,  215 
thousand,  86;   87  million,  16  thousand,  214;   93  million, 
18  thousand,  57 ;  246  million,  9  thousand,  456. 

59.  Find    the   sum    of    three    million,    eight    hundred 
twenty-four    thousand,    five    hundred    twenty-six;     forty 
million,   nineteen    thousand,   eight    hundred    twenty-five; 
eighty-six  million,  two   hundred  fifty-four  thousand,  two 
hundred ;  five  million,  five  thousand,  five. 

60.  What    is    the    sum  of    eighteen  million,   eighteen 
thousand,   eighteen;    thirty-five   million,   fifty-eight   thou- 
sand, two  hundred  seventy-eight ;  fifty-four  million,  seven 
hundred  forty-seven  thousand,  five  hundred  eighty-six  ? 

61.  What  is  the   sum  of  six  hundred  seven  thousand, 
two  hundred  eight;  eight  hundred  twenty-eight  thousand, 
nine  hundred ;  five  hundred  thousand,  fifteen  ? 

62.  Find  the  sum  of  twenty-eight  million,  one  hundred 
fifteen  thousand,  two  hundred ;  thirteen  million,  two  hun- 
dred   twenty-eight    thousand,    four;    nine    million,   eight 
thousand,  eight ;  ninety -nine  thousand,  nine  hundred  nine. 

63.  Add    twenty-five    million,     twenty-five     thousand, 
twenty-five;    forty   million,  four  hundred  thousand,  four 
hundred;  eighty  million,  eighty  thousand,  eighty;    eight 
hundred  thousand. 

64.  A   rolling  mill  in  Buffalo  turned  out  on  Monday 
1760  steel  rails ;  on  Tuesday,  1775 ;  on  Wednesday,  1809 ; 
on  Thursday,  1826 ;  and  on  Friday,  1919.     What  was  the 
total  number  made  in  the  five  days  ? 

65.  A  fruit-dealer  in  Pensacola  shipped  for  New  York 
in  one  week  2464  boxes  of  oranges,  1632  boxes  of  pine- 
apples, 1948  boxes  of  lemons,  and  850  boxes  of  cocoanuts. 
What  was  the  entire  number  of  boxes  of  fruit  shipped  ? 

STAND.   AR. — 3 


34  ADDITION. 

66.  In  the  month,  of  October  there  were  shipped  from 
New  Orleans  for  Philadelphia  during  the  first  week  14,573 
bales  of  cotton;  the  second  week,  17,849;  the  third  week, 
19,387;  and  during  the  last  week  of  the  month  23,879  bales. 
What  was  the  total  number  of  bales  sent  to  Philadelphia 
during  the  month  ? 

67.  Lake   Superior   covers   a  surface  of  32,290  square 
miles ;  Lake  Michigan,  23,903  square  miles ;  Lake  Huron, 
23,684;   Lake  Erie,  9493;   Lake  Ontario,  7654.     What  is 
the  entire  area  covered  by  these  lakes  ? 

68.  A  man  owns  five  horses.     The  first  is  worth  $250, 
the  second  $425,  the  third  $475,  the  fourth  as  much  as 
the  second  and  third,  and  the  fifth  as  much  as  the  first 
and  fourth.     What  is  the  value  of  the  five  horses  ? 

69.  An  orchard  contains  278  apple  trees,  and  an  equal 
number  of  pear  trees ;  354  peach  trees,  and  an  equal  num- 
ber of  plum  trees ;  and  117  cherry  trees.     How  many  trees 
are  there  in  the  orchard  ? 

70.  On  June  1,  a  lumber  firm  in  Portland  shipped  for 
Boston  73,452  shingles;  on  June  2,  56,280;  on  June  3,  4700; 
on  June  4,   87,950 ;  on  June  5,  4000  shingles  and  38,400 
laths ;  and  on  the  6th  of  June,  68,000  laths.     How  many  of 
each  were  shipped  in  the  six  days  ? 

71.  Mr.  George  Peabody  gave  to  the  poor  of  London 
$2,250,000;    to   the  town  of  Danvers,   $60,000;    to  the 
Grinnell  Arctic  Expedition,  $  10,000 ;  to  the  city  of  Balti- 
more, $1,000,000;    to  Phillips  Academy,  $25,000;  to  the 
Massachusetts   Historical   Society,   $20,000;    to    Harvard 
University,  $  150,000 ;  to  Yale  University,  $  150,000 ;  to  the 
Southwest,  $  1,500,000.     How  much  did  he  give  away  ? 

72.  A   drover  bought  horses  for  $3750,  and  cows  for 
$2875.     On  his  horses  he  gained  $976,  and  on  his  cows 


WRITTEN   EXERCISES.  35 

$673.     What  would  lie  have  received  for  both  if  he  had 
gained  $  500  more  than  he  did  ? 

73.  The  Missouri  Kiver,  to  its  junction  with  the  Missis- 
sippi, is  2908  miles  long;   the  Mississippi  proper  is  2616 
miles   long;   the   St.   Lawrence   is   2120  miles   long;   the 
Amazon  3596  miles  long.    What  is  the  combined  length  of 
these  rivers  ? 

74.  New  York  is  1405  miles  east  of  Omaha,  and  San 
Francisco  is  1864  miles  west  of  Omaha.     How  far  is  it 
from  New  York  to  San  Francisco  ? 

75.  A  gentleman  willed  his  property  to  his  wife,  three 
sons,  and  four   daughters;    to  each  of  his   daughters   he 
willed  $3869;   to  his  sons  each  $4781;   and  to  his  wife 
$  12,000.     How  much  was  his  property  ? 

76.  From  the  nail  works  at  Pittsburg  there  were  shipped 
7650  kegs  of  nails  on  Monday,  8640  kegs  on  Tuesday,  300 
kegs  more  on  Wednesday  than  on  Tuesday,  9850  kegs  on 
Thursday,  and  10,000  kegs  on  eacn  of  the  remaining  two 
days  of  the  week.     How  many  kegs  were  shipped  during 
the  week  ? 

77.  A  train  left  Rutland  with  seven  cars  loaded  with, 
marble,  as  follows :  The  first  car  had  on  it  36,725  pounds ; 
the  second,  36,850;  the  third,  37,200;  the  fourth,  37,650; 
the  fifth,  37,150 ;  the  sixth,  38,090 ;  and  the  seventh,  27,360 l 
pounds.     How  many  pounds  of  marble  were  there  on  the 
train  ? 

78.  If  a  button  manufactory  at  Lowell  made  in  one  day 
17,200   buttons,  in  another  18,560,  in  another  18,569,  in 
another  16,502,  and  in  another  18,250,  how  many  buttons 
were  made  in  those  five  days  ? 


36  ADDITION. 

79.  The  area  of  Maine  in  square  miles  is  33,040;  of  New 
Hampshire,  9305 ;    of  Vermont,  9565 ;    of   Massachusetts, 
8315 ;  of  Ehode  Island,  1250 ;  of  Connecticut,  4990.     What 
is  the  area  of  New  England  in  square  miles  ? 

80.  Mr.   A  deposited  in  the   First  National  Bank  of 
Albany,  N.Y.,  on   June   5,  1892,  $469.50;    on   June   10, 
$764.35;   on  June   12,   $320;   on  June  16,   $125.75;   on 
June  18,  $  673.85.     He  also  deposited  in  the  National  Park 
Bank  of  New  York  City,  on  June  22,  1892,  $3450.27;  and 
on  June  24,  $  1250.     How  much  did  he  deposit  in  each  of 
the  banks  ?     How  much  in  both  banks  ? 

81.  A  glass  factory  in  Newark  made  in  one  day  1760 
tumblers,  860  goblets,  2125  ounce  bottles,  1240  two-ounce 
bottles,  375  fruit  jars,  3600  glass  tubes,  and  1200  other 
articles  for  laboratory  purposes.     How  many  articles  were 
made  in  all  ? 

82.  A  grain  dealer  in  Chicago  shipped  3750  bushels  of 
wheat  and  4560  bushels  of  corn  to  Baltimore  one  week;  the 
next  week  4675  bushels  of  wheat  and  5000  bushels  of  corn ; 
and  the  week  following  6000  bushels  of  wheat  and  7180 
bushels  of  corn.     How  many  bushels  of  each  kind  did  he 
ship  ?     How  many  bushels  of  grain  ? 

83.  If  it  takes  16,718  bricks  to  build  a  dwelling  house, 
39,900  for  a  school  building,  and  50,000  for  a  church,  how 
many  bricks  will  be  required  for  the  three  buildings  ? 

84.  A  business   firm   sold  in  the  month  of  December 
$  6575  worth  of  goods  ;  a  second  firm  sold  $  7480  worth ;  a 
third,  $  7850 ;  a  fourth,  $8175 ;  a  fifth,  $  8262;  and  a  sixth 
firm  sold  $  9150  worth.     What  was  the  value  of  the  goods 
aold? 


WRITTEN  EXERCISES.  37 

85.  If  in  Hartford  there  were  made   during  the  first 
week  in  September  3980  clocks,  the  second  week  3986,  the 
third  week  4015,  and  the  fourth  week  4220,  how  many 
clocks  were  made  there  during  those  four  weeks  ? 

86.  Virginia  contains  42,450  square  miles ;   Tennessee, 
42,050  square  miles;  North  Carolina,  10,200  square  miles 
more  than  Tennessee;  and  Louisiana  6270  square  miles' 
more    than    Virginia;    Maryland    contains   12,210  square* 
miles.    How  many  square  miles  do  all  these  states  contain  ? 

87.  In  a  certain  town  there  were  manufactured  in  one 
week  5860  yards  of  broadcloth,  7970  yards  of  black  cheviot, 
9370  yards  of  muslin,  6250  yards  of  calico,  3600  yards  of 
gingham,  and  12,000  yards  of  various  other  woolen  and 
cotton  goods.     How  many  yards  were  manufactured  in  the 
entire  week  ? 

88.  A  cork  factory  made  16,150  corks  on  Monday,  17,050 
on  Tuesday,  17,364   on  Wednesday,  17,500  on  Thursday, 
18,008  on  Friday,  and  18,169  on  Saturday.     The  week  fol- 
lowing 4000  more  corks  were  made  than  during  the  pre- 
ceding week.     How  many  corks  were   made   in  the  two 
weeks  ? 

89.  On  October  23  a  vessel  arrived  at  Cincinnati  from 
Mobile  having  on  board  90  bales  of  cotton,  weighing  27,600 
pounds,  in  one   part  of  the  ship,  and  17,900   pounds   of 
cotton  in  another  part;  also  156  hogsheads  of  cane  sugar, 
5060  gallons  of  cane  molasses,  and  1260  gallons  of  sorghum 
molasses.     On  the   same  day  a  vessel  arrived  from  New 
Orleans,  with  45,760  pounds  of  cotton,  3600  gallons  of  cane 
molasses,  630  gallons  of  sorghum  molasses,  and  175  hogs- 
heads of  cane  sugar.     How  many  pounds  of  cotton,  how 
many   gallons   of  molasses,  and  how   many  hogsheads  of 
sugar  were  there  on  both  vessels  ? 


SUBTRACTION. 


45.    1.    How  many  books  are  left  when  3  books  are  taken 
from  6  books  ? 

2.  How  many  horses  are  left  when  3  horses  are  taken 
from  4  horses  ? 

3.  Henry  drew  5  pictures,  all   but  2   of   which   were 
pictures  of  birds.     How  many   pictures   of   birds   did  he 
make  ? 

4.  What  is  the  difference  between  4  cents  and  2  cents  ? 

5.  What  is  the  difference  between  5  and  3?     Between 
4  and  2  ? 

6.  What  have  you  been  doing  with  the  numbers  given 
above  ? 

7.  Take  3  from  4,  and  what  number  remains  ?     Add 
this   remainder  to  the  smaller  number,  and  tell  how  the 
sum  compares  with  the  larger  number. 

8.  Take   3  from  5,  and  what  number  remains  ?     Add 
this  remainder  to  the  smaller  number,  and  tell   how  the 
sum  compares  with  the  larger  number  ? 

9.  When  one  number  is  subtracted  from  another,  how 
does  the  sum  of  the  remainder  and  the  smaller  compare 
with  the  larger  ? 

10.   Why  can  we  not  express  the  difference  between  6 
apples  and  5  cents  ? 
38 


DRILL  EXERCISES.  39 

46.  The  process  of  finding  what  is  left  when  a  part  of  a 
number  is  taken  away  from  it,  or  of  finding  the  difference 
between  two  numbers  is  called  Subtraction. 

47.  The  number  from  which  another  is  to  be  subtracted 
is  called  the  Minuend. 

48.  The  number  to  be  subtracted  is  called  the  Subtrahend. 

49.  The   result  obtained  by   subtracting  is   called   the 
Remainder  or  Difference. 

50.  The  Sign  of  Subtraction  is  a  short  horizontal  line  — . 
It  is  called  minus.     When  it  is  placed  between  two  num- 
bers, it  shows  that  the  one  after  it  is  to  be  subtracted  from 
the  one  before  it. 

Thus,  9  —  5  is  read  9  minus  5,  and  means  that  5  is  to  be  subtracted 
from  9. 

51.  PRINCIPLES  :  —  1.   Only  like  numbers  can  be  subtracted. 
2.   The  sum  of  the  subtrahend  and  the  remainder  is  equal 

to  the  minuend. 

DRILL  EXERCISES. 

52.  Students  should  practice  these  exercises  daily  until 
they  can  tell  the  results  instantly. 

1.  9  +  ?  =  15      15-9  =  ?      8  +  ?  =  13  13-8  =  ? 

2.  7+?  =  17      17-7  =  ?       6  +  ?  =  15  15-6  =  ? 

3.  8  +  ?  =  19      19-8=?      5  +  ?  =  14  14-5  =  ? 

4.  4  +  ?  =  13       13-4  =  ?       9  +  ?  =  17  17-9  =  ? 

5.  7  +  ?  =  15      15-7=?       8+?  =  18  18-8=? 

6.  13-3        10-4          9-6          8-4  11-3 

7.  9-9        12-3        11-2          8-6          7-4 

8.  11-6        12-9        10-3        10-2        12-2 


40  SUBTRACTION. 

9.  10-7  9-8  11-9  9  —  7  8  —  2 

10.  9-3  8  —  7  9  —  2  7  —  6  10-9 

11.  7  —  2  6-5  7  —  3  6  —  4  5  —  2 

12.  6-3  8  —  3  7-5  5  —  3  8-5 

13.  6-2  3-2  4-3  4-2  5-4 

14.  17-4         18-6        18-5        15-8        14-4 

15.  17  — 6         16-8         12  —  6        13  —  9        18-9 

16.  11-5        16-4        19-9        13-6        12-7 

17.  13-5        17-8         12  —  8         10  —  5          9-5 

18.  10-8         16-6         12-5        14-6        12-4 

19.  18-7        11-7        14-2        18-8        13-7 

20.  17-5         14-9        11-8        15-4        19-7 

21.  14-8          9-4        11-4        14-3        15-2 

22.  15-5        16-7        16-9        15-3        13-2 

23.  10-6         16-5         14-7         19-6         16-3 

24.  Subtract  by  2's  from  24  to  0 ;  thus,  24,  22,  20, 18,  etc. 

25.  Subtract  by  3's  from  36  to  0.  From  34  to  1. 

26.  Subtract  by  4's  from  40  to  0.  From  43  to  3. 

27.  Subtract  by  5's  from  55  to  0.  From  57  to  2. 

ORAL  EXERCISES. 

53.   1.   If   I   have   12  cents,  and  spend  5   cents   for   a 
pencil,  how  many  cents  will  I  have  left  ?     12  —  5  =  ? 

2.  A  man  earns  $15  per  week,  but  spends  $10.     How 
much  has  he  left  at  the  end  of  the  week  ?     15  —  10  =  ? 

3.  Our  lesson  contained  11  problems,  of  which  I  solved 
all  but  3.     How  many  did  I  solve  ?     11  —  3  =  ? 

4.  A  hen  hatched  13  chicks,  but  5  of  them  died.     How 
manylived?     13-5=?     13-3  =  ?     13-4=? 


ORAL  EXERCISES.  41 

5.  Henry  found  that  his  toy  bank  contained  14  cents, 
but  8  cents  of  ihe  money  belonged  to  his  brother.     How 
much  belonged  to  Henry  ? 

6.  William   and  James   together   caught   15   fish.     If 
William  caught  7  of  them,  how  many  did  James  catch? 
15  -  7  =  ? 

7.  Mary  is  13  years  old,  and  her  brother  is  8.     How 
much  older  is  Mary  than  her  brother  ?     13  —  8  =  ? 

8.  Homer  was  away  from  home  2  weeks.     He  spent  5 
days  with  his  uncle,  and  the  rest  of  the  time  with  his  grand- 
father.    How  many  days  was  he  with  his  grandfather  ? 

9.  I  bought  an  orange  for  3  cents,  a  banana  for  2  cents, 
and  candy  for  5  cents.     How  much  did  I  have  left  after 
paying  for  all,  if  I  had  15  cents  to  begin  with  ? 

10.  A.  boy  earned  during  vacation  $15.     He  spent  $2 
for  books,   $3   for   clothing,   and   $5    for    other    things. 
How  much  had  he  left  ?     15-2-3-5=? 

11.  By  an  accident  a  boy  had  2  fingers  cut  from  his  right 
hand,  and  3  from  his  left  hand.     How  many  fingers  has  he 
left?     10-3-2=? 

12.  A  gentleman  who  had  7  horses  bought  6,  and  after- 
ward sold  at  one  time  4,  and  at  another  3.     How  many  had 
he  left  ? 

13.  If  a  man  earns  $  19  per  week  and  spends  $  10,  how 
much  does  he  save  ? 

14.  James  received  18  marbles  at  Christmas,  of  which  8 
were  glass  and  the  rest  clay.     How  many  were  clay  ? 

15.  How  much  will  I  have  left  if  I  have  15  cents  and 
spend  8  cents  ? 

16.  A  girl  had  to  knit  15  rows  to  finish  her  work.    After 
she  had  knit  6  rows  how  many  had  she  still  to  do  ? 


42  SUBTRACTION. 

17.  A  park  had  19  elm  trees  in  it,  of  which  9  were  small 
and  the  rest  large.     How  many  were  large.? 

18.  Alice  has  18  pins,  and  Ella  has  7.     How  many  more 
than  Ella  has  Alice  ? 

19.  A  man  earned  $  17  and  spent  $  8.     How  many  dollars 
had  he  left  ? 

20.  Harry  had  14  cents  given  him  by  two  boys.     One 
gave  him  6  cents.     How  much  did  the  other  give  ? 

21.  John  earned  $  7,  and  his  father  gave  him  enough  to 
make  his  money  $  16.     How  much  did  his  father  give  him  ? 

22.  A  newsboy  bought  papers  for  9  cents,  and  sold  them 
for  17  cents.     How  much  did  he  gain  ? 

23.  If  I  have  13  cents  and  spend  6  cents  for  a  ball  and 
3  cents  for  a  pencil,  how  much  will  I  have  left  ? 

24.  There  were  15  birds  on  a  tree,  and  4  of  them  flew 
away.     How  many  remained  on  the  tree  ? 

25.  James  had  14  apples  and  ate  5  of  them.     How  many 
had  he  left  ? 

26.  Charles  is  17  years  old,  and  his  brother  is  6  years 
younger.     How  old  is  his  brother  ? 

27.  Joseph  had  8  cents,  and  his  father  gave  him  5  cents. 
He  afterwards  spent  7  cents.     How  much  money  had  he 
left? 

28.  Ada  picked  13  quarts  of  cherries,  and  Emma  7  quarts. 
How  many  more  quarts  must  Emma  pick  so  that  the  amount 
she  picked  will  equal  what  Ada  picked  ? 

29.  A  man  had  16  horses  and  sold  9  of  them.     How 
many  horses  did  he  keep  ? 

30.  From  a  pile  of  wood  containing  17  cords,  a  farmer 
sold  at  one  time  5  cords,  and  at  another  time  6  cords.     How 
many  cords  remained  unsold  ? 


WRITTEN   EXERCISES.  43 

31.  A  watch  cost  $15,  and  was  sold  for  $6  less  than 
cost.     For  how  much  was  it  sold  ? 

32.  From  a  bin  containing  19  bushels  of  wheat,  5  bushels 
were  taken  for  seed  and  7  bushels  were  sold.     How  many 
bushels  remained  in  the  bin  ? 

33.  A  farmer  having  9  cows,  bought  at  one  time  3,  and 
at  another  time  5,  and  afterwards  sold  10.     How  many  cows 
had  he  then  ? 

WRITTEN    EXERCISES. 
54.   1.   From  796  subtract  343. 

Minuend,       796          EXPLANATION. — For  convenience  the  less  num- 
Subtrahend    343     ber  is  written  under  the  greater,  units  under  units, 

tens  under  tens,  etc. 

Remainder,    453          Each  order  of  units  of  the  subtrahend  is  then 
subtracted  separately  from  the  same  order  in  the 
minuend,  and  the  remainders  are  written  beneath. 

PROOF.  —  453,  the  remainder,  plus  343,  the  subtrahend,  equals  796, 
the  minuend.     Hence  the  result  is  correct  (Prin.  2). 


5.  6. 

584  295 

420  173 


Copy,  subtract,  and  prove  : 

2. 

3. 

4. 

734 
412 

685 
541 

767 
634 

7. 

8. 

9. 

7596 
3152 

8493 
6271 

5694 
4253 

12. 

13. 

14. 

$38.76 
15.24 

$67.58 
33.24 

$93.85 
61.50 

10.  11. 

6897      6843 
5385      4122 


44  SUBTRACTION. 


17. 

18. 

19. 

20. 

21. 

$85.39 
54.08 

$  76.83 
66.02 

23. 

$69.84 
55.41 

$63.73 
31.61 

$91.87 
50.43 

22. 

24. 

25. 

26. 

95684 
71271 

86431 
73210 

81396 
40285 

79265 
41132 

51869 
20714 

27. 

28. 

29. 

30. 

31. 

98316 
71004 

83468 
41215 

68793 
54251 

88416 
42304 

48319 
24207 

32.  A  house  was  bought  for  $1639,  and  sold  for  $1859. 
What  was  the  gain  ? 

33.  A  man  bought  a  farm  for  $8768,  and  sold  it  for 
$  6424.     What  was  the  loss  ? 

34.  Mr.  A  traveled  2168  miles,  and  Mr.  B  traveled  1145 
miles.     How  much  farther  did  Mr.  A  travel  than  Mr.  B  ? 

35.  A  drover  having  1836   sheep   sold   1220   of  them. 
How  many  had  he  left  ? 

36.  I  bought  a  house  for  $3425,  and  sold  it  for  $4538. 
How  much  did  I  gain  ? 

37.  A  farmer  who  raised  2560  bushels  of  corn  sold  all 
but  350  bushels.     How  much  did  he  sell  ? 

38.  Two  men  together  own  5656  acres  of  land.     If  the 
one  owns  2535  acres,  how  many  acres  does  the  other  own  ? 

39.  A  man  bought  a  horse,  a  buggy,  and  a  harness  for 
$  278.75.  The  buggy  and  harness  cost  him  $  136.50.  What 
did  he  pay  for  the  horse  ? 


ORAL   EXERCISES.  45 

40.  A  man's  income  for  one  year  was  $2568.48,  and  his 
expenses  were  $  1445.23.     How  much  did  he  save  ? 

41.  A  builder  contracted  to  build  a  house  for  $2885. 
The  expenses   for  material  and  labor   were  $  2552.     How 
much  were  the  profits  ? 

42.  A  grain  dealer  bought  in  one  week  38,547  bushels  of 
wheat,  and  sold  25,336  bushels.     How  much  more  did  he 
buy  than  he  sold  ? 

ORAL    EXERCISES. 

55.   1.   A  jeweler  bought  a  watch  for  $18,  and  sold  it 
for  $  25.     How  much  did  he  gain  ? 

2.  A  man  proposed  to  walk  30  miles,  but  after  he  had 
walked  19  •  miles  he  stopped.     How  many  more  miles  must 
he  travel  to  complete  the  distance  ? 

3.  A  lady  made  purchases  to  the  amount  of  $17,  which 
she  paid  for  with  a  twenty-dollar  bill.     How  much  change 
did  she  receive  ? 

4.  A  grocer  purchased  tea  for  $  32,  and  sold  it  for  $  40. 
How  much  was  his  gain  ? 

5.  Two  trains  left  New  Orleans  at  the  same  time,  one 
running  31  miles  per  hour  and  the  other  24  miles  per  hour. 
What  was  the  difference  in  the  rate  of  speed  ? 

6.  The  Empire  State  express  train  runs  about  53  miles 
per  hour,  and  other  fast  express  trains  about  40  miles  per 
hour.     How  much  faster  does  the  Empire  State  train  run  ? 

7.  A  boy  earned  50  cents  per  day,  but  was  obliged  to 
pay  15  cents  for  a  lunch  and  10  cents  for  car  fare.     How 
much  did  he  save  daily  ? 

8.  The  receipts  of  an  entertainment  were  $35,  and  the 
expenses  $  26.    What  were  the  profits  ? 


46  SUBTRACTION. 

9.  A  lady  purchased  a  table  for  $  7  and  a  chair  for  $  13. 
How  much  change  should  she  receive  if  she  gave  the 
merchant  a  fifty-dollar  bill  ? 

10.  A  three-story  house  was  42  feet  high.      The  first 
story  was  15  feet,  and  the  second  14  feet.     How  high  was 
the  third  story  ? 

11.  A  boy  read  47  books  during  the  year,  of  which  20 
were  books  of  travel,  12  histories,  and  the  rest  stories. 
How  many  stories  did  he  read  ? 

12.  9  +  7-3  +  2-54-4-5  +  7-4-2-5=? 

13.  8  +  3  +  2-5  +  2-4  +  7-3  +  5-6  +  2=? 

14.  8  +  2-5  +  9-3  +  4-5  +  7  +  3-5-4=? 

15.  5  +  6  +  2-5  +  4-6  +  9-3-2+4-2=? 

16.  3  +  9-3  +  4-2-4  +  8  +  2-6  +  3-5=? 

17.  4  +  6-5+8-3  +  7-6-2  +  9  +  1-4=? 

18.  7  —  3  +  7  +  3  +  5  —  6  +  9  —  3  +  5  —  6-4=? 

19.  6  +  5-4  +  4-2  +  5-7  +  4  +  2-5-2=? 

20.  8-2  +  9-7  +  6  +  3-5  +  2-7  +  5  +  6=? 

21.  5  +  4-3  +  7-2-5  +  6  +  8-3-5  +  4=? 

22.  Subtract  by    6's  from    66  to  0.     From    61  to  1. 

23.  Subtract  by    7's  from    63  to  0.     From    65  to  2. 

24.  Subtract  by    8's  from    56  to  0.     From    74  to  2. 

25.  Subtract  by    9's  from    72  to  0.     From    75  to  3. 

26.  Subtract  by  10's  from    95  to  5.     From    87  to  7. 

27.  Subtract  by  20's  from  106  to  6.     From  145  to  5. 

28.  Subtract  6  from  34,  44,  54,  64,  74,  84. 

29.  Subtract  7  from  32,  42,  52,  62,  72,  82. 

30.  Subtract  8  from  35,  45,  55,  65,  75,  85. 


WRITTEN  EXERCISES.  47 


WRITTEN   EXERCISES. 

56.   1.   From  925  subtract  476. 

EXPLANATION.  —  The  subtrahend  is  written  under  the  minuend,  and 
we  begin  at  the  right  to  subtract. 

Since  6  units  cannot  be  subtracted  from  5  units,  one  of  the 
925  tens,  which  is  equal  to  10  units,  is  united  with  the  5  units, 
476  making  15  units.  6  units  from  15  units  leave  9  units,  which 

are  written  under  the  units. 

449  Since  one  of  the  tens  was  united  with  the  units,  there  is  but 
1  ten  left.  Because  7  tens  cannot  be  subtracted  from  1  ten,  1 
hundred,  which  is  equal  to  10  tens,  is  united  with  the  1  ten,  making 
11  tens.  7  tens  from  11  tens  leave  4  tens,  which  are  written  under 
the  tens. 

Since  one  of  the  hundreds  was  written  with  the  tens,  there  are  but 
8  hundreds  left.  4  hundreds  from  8  hundreds  leave  4  hundreds,  which 
are  written  under  the  hundreds.  Hence  the  remainder  is  449. 

PROOF.  —  449,  the  remainder,  plus  476,  the  subtrahend,  equals  925, 
the  minuend.  Hence  the  result  is  correct  (Prin.  2). 

RULE.  —  Write  the  subtrahend  under  the  minuend,  units 
under  units,  tens  under  tens,  etc. 

Begin  at  the  right  and  subtract  each  figure  of  the  subtrahend 
from  the  corresponding  figure  of  the  'minuend,  writing  the  re- 
sult beneath. 

If  a  figure  in  the  minuend  has  a  less  value  than  the  corre- 
sponding figure  in  the  subtrahend,  increase  the  former  by  ten, 
and  subtract;  then  diminish  by  one  the  units  of  the  next  higher 
order  in  the  minuend,  and  subtract  as  before. 

PROOF.  — Add  together  the  remainder  and  subtrahend.  If 
the  result  is  equal  to  the  minuend,  the  work  is  correct. 

Subtract  and  prove : 

2.  913-426.            5.  913-746.  8.  592-176. 

3.  835  —  341.            6.  583  —  279.  9.  735  —  444. 

4.  736-453.            7.  468-349.  10.  596-288. 


SUBTRACTION. 


11.  732-456. 

12.  594-427. 

13.  635-387. 

14.  925-443. 

15.  876-687. 

16.  319-127. 

17.  486-349. 


18.  782-458. 

19.  854-527. 

20.  834-456. 

21.  318-129. 

22.  893-467. 

23.  752-556. 

24.  816-724. 


25.  931-588. 

26.  714-329. 

27.  531-423. 

28.  685  —  395. 

29.  419-199. 

30.  327-159. 

31.  743-358. 


32.  39456 

33.  48317 

34.  87593 

35.  81364 

36.  73586 

37.  39271 

38.  98375 

39.  83125 

40.  63259 

50.  $615.29 

51.  $732.84 

52.  $459.97 

53.  $159.13 

54.  $843.25 

55.  $917.36 

56.  $593.18 

57.  $691.23 


31567. 
27592. 
52869. 
68537. 
49288. 
14683. 
45792. 
72165. 
42199. 


$492.36. 
$537.39. 
$263.18. 
$137.29. 
$  391.65. 
$427.58. 
$505.09. 
$319.47. 


41.  49836 

42.  84391 

43.  73186 

44.  49315- 

45.  37926 

46.  72853 

47.  91835 

48.  42931 

49.  52361 


58.  $392.18 

59.  $576.88 

60.  $813.95 

61.  $927.86 

62.  $593.70 

63.  $315.91 

64.  $296.30 

65.  $483.35 


31849. 
43875. 
38592. 
18674. 
18395. 
41687. 
84635. 
28724. 
23854. 

$237.43. 
$499.90. 
$358.16. 
$423.58. 
$345.96. 
$256.19. 
$235.84. 
$219.35. 


WRITTEN  EXERCISES.  49 

66.  From  9000  subtract  7685. 

89910  EXPLANATION.  —  Since  6  units  cannot  be  subtracted  from 
9000  0  units,  and  since  there  are  no  tens  nor  hundreds,  1  thousand 
7gg5  must  be  changed  into  hundreds,  leaving  8  thousand  ;  1  of  the 

hundreds  must  be  changed  into  tens,  leaving  9  hundreds  and 

1315    1  of  the  tens  into  units,  leaving  9  tens.     The  expression  8 
thousands,  9  hundreds,  9  tens,  and  10  units  is  thus  equivalent 
to  the  minuend,  from  which  the  units  of  the  subtrahend  can  be 
readily  subtracted. 

Subtract  and  prove : 

67.  50000   -38517.  80.  $39600.85-$   1915.68. 

68.  60000    -29365.  81.  $  58400.00 -$  29318.54. 

69.  55000   -51093.  82.  $38600.05-$   3743.08. 

70.  39000   -28739.  83.  $50000.00-$    1830.05. 

71.  80000    —65004.  84.  $39000.65  —  $     937.97. 

72.  30040    -18391.  85.  $  30040.08 -$  19275.09. 

73.  70101    -43217.  86.  $  70410.00 -$  45200.18. 

74.  99003   -45009.  87.  $  60060.60 -$  39123.54. 

75.  834760-83290.  88.  9138700-   23047. 

76.  410506-23837.  89.  8004040-183079. 

77.  175004-23516.  90.  6700880-   58369. 

78.  393400  —  16042.  91.  5970008—      4999. 

79.  913043—   4009.  92.  3002250  —  210596. 

93.  A   borrowed   of  B    $6450,   and   paid  back   $3740. 
How  much  does  he  still  owe  ? 

94.  A  merchant  bought  a  quantity  of  goods  for  $15,125, 
and  sold  them  for  $  17,015.     What  was  the  gain  ? 

95.  The  sum  of  two  numbers  is  9416,  and  the  greater 
is  6809.    What  is  the  less  number  ? 

96.  The  year  1891  was  399  years  after  the  discovery  of 
America  by  Columbus.     In  what  year  did  that  event  take 
place  ? 

97.  B  bought  some  goods  which   he   sold  for  $11,325, 
and  thereby  gained  $  2150.     How  much  did  they  cost  him  ? 

STAND.    AR.  —  4 


50  SUBTRACTION. 

98.  I  bought  a  horse  for  $  325  and  a  cow  for  $  150.     I 
sold  the  horse  for  $410,  and  the  cow  for   $216.     How 
much  did  I  gain  by  the  sale  ? 

99.  A  merchant  deposited  in  a  bank  on  Monday  $584  ; 
on  Tuesday,    $759;    and   on  Wednesday,    $327.     During 
this  time  he  drew  out  $  987.     How  much  did  his  deposits 
exceed  what  he  drew  out  ? 

100.  A  man  bought  16,750  bricks,  and  then  sold  B  and 
€  each  4926.     How  many  had  he  left  ? 

101.  In  an  army  of  7569  men,  388  were  killed,  432  were 
wounded,  and  273  deserted.    How  many  remained  for  duty  ? 

102.  A  man  bought  a  farm  for  $7850.     He  expended 
$2169  for  improvements,  paid  $97  for  taxes,  and  then  sold 
it  for  $  10,650.     Did  he  gain  or  lose,  and  how  much  ? 

103.  A  man  left  $  3450  to  his  son,  $  2765  to  his  daughter, 
and  the  remainder  to  his  wife.     How  much  did  his  wife 
receive  if  the  fortune  was  $  20,000  ? 

104.  If  the   distance   of  the  moon   from   the   earth  is 
240,000  miles,  and  that  of  the  sun  95,000,000,  how  much 
farther  is  it  to  the  sun  than  to  the  moon  ? 

105.  On  Monday  morning  a  bank  had  on  hand  $2862. 
During  the  day  $  1831  were  deposited,  and  $  2172  drawn 
out ;  on  Tuesday,  $  3126  were  deposited,  and  $  1954  drawn 
out.    How  many  dollars  were  on  hand  Wednesday  morning  ? 

106.  B  had  $12,000;  but   after   paying   his  debts  and 
giving  away  $3105,  he  had  remaining  only  $7000.     What 
was  the  amount  of  his  debts  ? 

107.  A  had  $425,  B  had  $160  more  than  A,  and  C  had 
as  much  as  A  and  B  together.     How  much  had  C  ? 


WRITTEN   EXERCISES.  51 

108.  A  merchant  bought   500  yards  of  silk  for  $375, 
3500  yards  of  muslin  for  $  175,  and  600  yards  of  linen  for 
$  235 ;  he  sold  the  whole  for  $  1000.   How  much  did  he  gain  ? 

109.  An  estate  of  $12,350  was  divided  among  a  widow 
and  two  children.    The  widow's  share  was  $  6175,  the  son's, 
$2390  less  than  the  widow's,   and  the   rest   fell  to   the 
daughter.     What  was  the  daughter's  share  ? 

110.  A  drover  bought  300  horses  for  $32,150,  and  150 
cows  for  $4265,  and  sold  them  all  for  $  37,000.     What  was 
the  gain  ? 

111.  Mr.  E  bought  two  farms.     For  one  he  paid  $4560, 
and  for  the  other  $6000.      He   spent  on  each  $537  for 
improvements,  and  paid  taxes  which  amounted   to   $78. 
He  sold  both  farms  for  $  12,450.     Did  he  gain  or  lose  on 
the  sale,  and  how  much  ? 

112.  If  a  ship  was  bought  for  $43,650,  and  sold  for 
$45,000,  what  was  the  gain? 

113.  A  gentleman  gave  $13,465  for  a  house  and  some 
tand.     The  house  alone  was  worth  $  8978.    What  was  the 
value  of  the  land  ? 

114.  If  two  candidates  for  office  received  in  the  aggre- 
gate 93,565  votes,  and  the  successful  one  had  47,659  votes, 
how  many  did  the  other  have  ? 

115.  A  lumberman,  having  632,000  feet  of  boards,  sold 
328,582  feet  of  them.     How  many  feet  remained  unsold  ? 

116.  A    man    is    worth    $16,425,   of  which   $3750   is 
invested  in  bank  stock,  $  2746  in  mortgages,  and  the  rest  in 
land.     How  much  has  he  invested  in  land  ? 


MULTIPLICATION. 


57.    1.    How  many  cents  are  there  in  3  two-cent  pieces  ? 

2.  How  many  blocks  are  there  in  3  piles  containing  3 
blocks  each  ? 

3.  How  many  arc  two  4's  ?     Two  3's  ?     Three  3's  ? 

4.  What  have  you  been  doing  with  the  numbers  given 
above  ? 

5.  When  numbers  are  used  without  reference  to  any 
particular  thing,  what  name  is  given  to  them  ?     Abstract 
Numbers. 

6.  What  name  is  given  to  numbers  used  in  connection 
with  some  thing  ?     Concrete  Numbers. 

7.  How  many  trees  are  3  times  3  trees  ?     What  is  taken 
3  times  in  this  example  ? 

8.  How  many  ponies  are  4  times  3  ponies  ?    What  is 
taken  4  times  in  this  example  ? 

9.  How  many  cents  are  3  times  4  cents  ?    What  is  taken 
3  times  in  this  example  ? 

10.  In  each  example,  is  the  number  in  the  answer  like 
the  number  taken  or  like  the  number  which  tells  how  many 
times  the  number  is  taken  ? 

11.  Is  the  number  which  tells  how  many  times  the  other 
number  is  taken  concrete  or  abstract  ? 

12.  How  does  3  times  2  compare  with  2  times  3  ?  4  times 
2  with  2  times  4  ?  3  times  4  with  4  times  3  ? 

52 


DEFINITIONS.  53 

58.  The  process  of  taking  one  number  as  many  times  as 
there  are  units  in  another  is  called  Multiplication ;  or, 

Multiplication  is  a  short  process  of  adding  equal  numbers. 

59.  The  number  taken  or  multiplied  is  called  the  Multi- 
plicand. 

60.  The  number  showing  how  many  times  the  multipli- 
cand is  taken  is  called  the  Multiplier. 

61.  The  result  obtained  by  multiplying  is   called  the 
Product. 

62.  The  multiplicand  and  multiplier  are  called  the  Factors 
of  the  Product. 

63.  The  Sign  of  Multiplication  is  an  oblique  cross  x .     It 
is  read  multiplied  by  when  the  multiplicand  precedes  it  and 
times  when  the  multiplier  precedes  it. 

Thus,  4  x  3  is  read  4  multiplied  by  3  when  4  is  the  multiplicand, 
but  it  is  read  4  times  3  when  4  is  the  multiplier. 

64.  A  number  used  without  reference  to  any  particular 
thing  is  called  an  Abstract  Number. 

Thus,  4,  7,  9,  etc.,  are  called  abstract  numbers. 

65.  A  number  used  in  connection  with   some   thing  is 
called  a  Concrete  Number. 

Thus,  4  books,  7  days,  9  dollars,  are  concrete  numbers. 

66.  PRINCIPLES.  —  1.    The  multiplier  must  be  regarded  as 
an  abstract  number. 

2.  TJie  multiplicand  and  product  must  be  like  numbers. 

3.  Either  factor  may  be  used  as  multiplier  or  multiplicand 
when  both  are  abstract. 

In  practice,  for  convenience,  the  smaller  number  is  generally  used 
as  the  multiplier. 


54 


MULTIPLICATION. 


MULTIPLICATION   TABLE. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 
14 

18 
21 

24 

28 

30 
35 

36 
42 

42 

~49~ 

48 
~56~ 

54 

60 

66 

72 
84 

7 

63 

72 

70 

~8o~ 

77 

8 

16 

24 

32 

40 

48 

56 

64 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 
108 

110 
120 

121 
132 

132 

12 

24 

36 

48 

60 

72 

84 

96 

144 

EXPLANATION.  — The  numbers  in  the  left-hand  column  may  be 
regarded  as  the  multipliers,  and  the  numbers  across  the  top  as  the 
multiplicands.  The  products  will  be  found  in  the  horizontal  columns 
opposite  the  multipliers. 

Thus,  2  ones  are  2  ;  2  twos  are  4  ;  2  threes  are  6  ;  2  fours  are  8,  etc. 

The  order  may  be  changed  so  that  the  numbers  in  the  upper  hori- 
zontal line  may  be  regarded  as  the  multipliers,  and  the  numbers  on 
the  left  as  the  multiplicands.  The  products  will  then  be  found  in  col- 
umns under  the  multiplier. 

DRILL    EXERCISES. 


67. 

Find 

the 

products 

of: 

8x 

4. 

5 

xll. 

12 

X 

6. 

7 

X 

5. 

2 

X    7. 

5x 

3. 

12 

X    7. 

11 

X 

5. 

2 

X 

6. 

3 

x    4. 

6x 

8. 

9 

X    9. 

4 

X 

9. 

3 

X 

5. 

2 

x    4. 

7x 

2. 

8 

Xl2. 

5 

X 

5. 

4 

X 

2. 

12 

X    3. 

9x 

8. 

3 

X    3. 

7 

X 

6. 

5 

X 

2. 

9 

X    4. 

8x 

7. 

4 

XlO. 

9 

X 

7. 

6 

X 

9. 

10 

Xl2. 

3x 

9. 

5 

X    7. 

10 

X 

8. 

7 

X 

11. 

11 

x    4. 

4x 

6. 

6 

X    5. 

11 

X 

10. 

7 

X 

3. 

2 

X    3. 

5x 

6. 

11 

X    7. 

2 

X 

11. 

2 

X 

9. 

8 

X    6. 

ORAL  EXERCISES.  55 


3x    7. 

12  x    9. 

9x    2. 

4x    4. 

6x7. 

4x    8. 

7.x    8. 

5x    9. 

6x    3. 

12  x    8. 

5x12. 

9x6. 

7x4. 

10  x  11. 

10  x    4. 

6x11. 

10  x  10. 

9x11. 

8x11. 

8x10. 

7x10. 

11  X  11. 

10  x    7. 

7x12. 

4x7. 

8x9. 

12  x  12. 

8x8. 

9x10. 

10  x    3. 

9x    5. 

7x    7. 

4x    5. 

10  x    6. 

12  x    2. 

5x    8. 

8x5. 

5x10. 

11  X    9. 

4x12. 

3x10. 

2x8. 

6x12. 

12  x  11. 

3x2. 

4x    3. 

2x12. 

7x9. 

2x5. 

11  X    3. 

3x11. 

9x3. 

6x10. 

3x8. 

11  X    2. 

2x10. 

3x12. 

6x6. 

6x2. 

11  X    6. 

5x4. 

8x3. 

12  x    5. 

10  x    5. 

12  x  10. 

10  x    9. 

3x6. 

8x2. 

11  x  12. 

10  x    2. 

11  X    8. 

6x4. 

9x12. 

12  x    4. 

4x11. 

ORAL   EXERCISES. 
68.   1.  At  8  cents  each,  what  will  5  pencils  cost  ? 

2.  What  will  4  pairs  of  shoes  cost  at  $  6  a  pair  ? 

3.  A  boy  received  $4  per  week  for  his  wages.      How 
much  did  he  earn  in  8  weeks  ?    8  times  4  =  ? 

4.  How  far  can  a  man  walk  in  9  hours,  if  he  walks  4 
miles  per  hour  ?     9  times  4  =  ? 

5.  An  orchard  contained  8  rows  of  trees,  and  there  were 
7  trees  in  each  row.     How  many  trees  were  there  ? 

6.  A  girl  attended  school  every  school-day  for  9  weeks. 
How  many  days  was  she  at  school  ? 

7.  There  are  8  pints  in  a  gallon.     How  many  pints  are 
there  in  8  gallons  ?     8x8  =  ? 

8.  A  teacher  assigned  as  a  lesson  9  written  problems 
each  day  for  6  days.    How  many  did  she  assign  altogether  ? 

9.  How  much  must  I  pay  for  9  quires  of  paper  at  8  cents 
per  quire  ?     9  times  8  =  ? 


56  MULTIPLICATION. 

10.  A  contribution  for  a  poor   man  contained  9  five- 
cent  pieces.    How  much,  was  that  ? 

11.  Most  children  are  in  school  6  hours  per  day.     How 
many  hours  are  they  in  school  per  week  ? 

12.  Five  good  long  paces  are  a  rod  in  length.     How 
many  paces  are  there  in  7  rods  ? 

13.  The  railway  company  offered  excursion  tickets  to 
Niagara  Falls  and  return  at  $5  each.     How  much  would 
they  cost  for  a  party  of  8  children  ? 

14.  Six   boys   are  hoeing  corn.      If  each  hoes  4  rows 
every  half-day,  how  many  rows  will  they  all  hoe  in  5  days  ? 

15.  There  are  5  roses  on  a  branch.     How  many  roses 
would  there   be   on  8   such   branches  ?      How  many  are 
eight  5's  ? 

16.  When  flour   is  worth   $6   per  barrel  what  will   9 
barrels  cost? 

17.  What  is  the  cost  of  6  towels  at  8  cents  apiece  ? 

18.  At  9  cents  a  pound  what  is  the  cost  of  8  pounds  of 
twine  ? 

19.  A  set  of  table  knives  is  6.     How  many  knives  are 
there  in  8  sets  ?     • 

20.  A  sheet  of  note  paper  has  4  pages.      How  many 
pages  do  10  sheets  have  ?     8  sheets  ?     6  sheets  ? 

21.  Asa  earned  $9  per  week.    *How  much  did  he  earn 
in  7  weeks  ? 

22.  If  8  times  around  a  certain  course  is  just  a  mile, 
how  many  times  around  it  must  I  go  to  walk  8  miles  ? 

23.  Seven  boys   went  fishing.      James   caught   7   fish : 
if  they  had  each  caught  as  many,  how  many  would  all  have 
caught  ? 


WRITTEN  EXERCISES.  57 

24.  A  window  composed  of  small  panes  of  glass  had  10 
rows   containing  7  in  each.  row.     How   many  panes  were 
there  in  the  window  ? 

25.  How  many  marbles  have  8  boys,  if  each  boy  has  9 
marbles  ? 

26.  At  the  rate  of  6  miles  an  hour,  how  far  will  a  person 
travel  in  4  hours  ? 

27.  If  a  boy  writes  5  words  in  a  minute,  how  many 
words  can  he  write  in  9  minutes  ? 

28.  There  are  7  days  in  a  week.     How  many  days  are 
there  in  7  weeks  ? 

29.  There  are  8  quarts  in  a  peck.     How  many  quarts 
are  there  in  4  pecks,  or  a  bushel  ? 

30.  John  is  6  years  old,  and  his  father  is  5  times  as  old. 
How  old  is  his  father  ? 

31.  Maggie  made  a  quilt  with  8  rows  of  squares  in  it, 
and  8  squares  in  each  row.     How  many  squares  are  there 
in  the  quilt  ? 

32.  What  will  8  quarts  of  berries  cost  at  7  cents  a  quart  ? 

33.  If  a  man  can   earn   $6   in  one  week,  how   many 
dollars  can  he  earn  in  6  weeks  ? 

34.  I  bought  9  sheep  for  $  7  each.      How  much  did  I 
pay  for  them  ? 

WRITTEN  EXERCISES. 
69.   1.   How  many  are  5  times  364? 

EXPLANATION.  —  For  convenience,  the  multiplier  is  written  under 
the  multiplicand,  and  we  begin  at  the  right  to  multiply. 

5  times  4  units  are  20  units,  or  2  tens  and  0  units. 
Multi  licand  364    Tlie  ®  is  written  m  un^8'  place  in  the  product,  and 

the  2  tens  are  reserved  to  add  to  the  tens. 

Multiplier  5  times  6  tens  are  30  tens,  plus  2  tens  reserved, 

Product  1820  are  ^2  tens'  or  3  hundreds  and  2  tens.  The  2  is 
written  in  tens'  place  in  the  product,  and  the  3  hun- 
dreds are  reserved  to  add  to  the  hundreds. 


58 


MULTIPLICATION. 


5  times  3  hundreds  are  15  hundreds,  plus  3  hundreds  reserved,  are 
18  hundreds,  or  1  thousand  and  8  hundreds,  which  are  written  in 
thousands'  and  hundreds'  places  in  the  product. 

Hence  the  product  is  1820. 

Find  the  products  of  the  following : 

2.  3  x  342.  16.  6734  x 

3.  4  x  516.  17.  5921  x 

4.  5x378.  18.  6804  x 

5.  4  x  427.  19.  5387  x 

6.  3x543.  20.  2956  x 

7.  5x685.  21.  8543  x 

8.  3x379.  22.  7916  x 

9.  6x428.  23.  4438  x 

10.  4x385.          24.   7985  x 

11.  5  x  619.  25.   9983  x 

12.  11  x  384.  26.  8974  x 

13.  12  x  527.  27.  9376  x  11. 

14.  11  x  627.  28.  8437  x  12. 

15.  12  x  598.  29.  9546  x  11. 

44.  What  will  9  pairs  of  boots  cost  at  $ 5.25  a  pair? 

45.  A  machinist  earned  $18.75  per  week.     How  much 
did  he  earn  in  6  weeks  ? 

46.  At  an  auction  sale  a  man  bought  7  sewing  machines 
at  $  13.45  apiece.     How  much  did  they  all  cost  him  ? 

47.  A  builder  bought  8  loads  of  lumber  at  an  average 
price  of  $  16.35  per  load.     How  much  did  the  lumber  cost 
him? 

48.  At  $  19.65  cents  each,  what  would  7  stoves  cost  ? 

49.  A  man  bought  5  barrels  of  flour  at  $5.75  per  barrel. 
How  much  did  the  flour  cost  him  ? 


4. 

30. 

4  x  36895. 

6. 

31. 

7  x  29756. 

5. 

32. 

8  x  38946. 

7. 

33. 

9  x  69138. 

4. 

34. 

5  x  56847. 

6. 

35. 

8  x  38954. 

8. 

36. 

9  x  41689. 

7. 

37. 

7  x  39125. 

5. 

38. 

8  x  86438. 

6. 

39. 

9  x  60854. 

7. 

40. 

7  x  70685. 

1. 

41. 

5  x  71650. 

2. 

42. 

9  x  81459. 

1. 

43. 

8  x  92368. 

WRITTEN  EXERCISES.  59 

50.  A  drover  bought  6  head  of  cattle  at  an  average  cost 
of  $  37.45  a  head.     How  much  did  he  pay  for  them  ? 

51.  Mr.  A  bought  8  acres  of  land  at  $45.75  per  acre. 
How  much  did  the  land  cost  him  ? 

52.  If  a  firkin  of  butter  costs  $  13.85,  what  must  be  paid 
for  5  firkins  ? 

53.  There  are  5280  feet  in  a  mile.     How  many  feet  are 
there  in  6  miles  ? 

54.  There   are  3600   seconds   in   1   hour.     How   many 
seconds  are  there  in  8  hours  ? 

55.  What  will  7  tons  of  railroad  iron  cost  at  $64.25  a 
ton? 

56.  A  merchant  sold  9  bales  of  cotton  at  $37.60  a  bale. 
How  much  did  he  receive  for  it  ? 

57.  At  a  public  sale  8  tons  of  hay  were  sold  for  $7.85 
per  ton.     How  much  was  paid  for  the  hay  ? 

58.  A  party  of  6  persons  hired  a  schooner,  paying  $3.75 
apiece  for  the  use  of  it.     How  much  did  all  pay  ? 

59.  What  will  be  the  cost  of  7  organs,  if  each  organ  is 
worth  $65.75? 

60.  At  $  75.35  each,  what  must  be  paid  for  8  wagons  ? 

61.  At  an  average  weight  of  1267  pounds  each,  what  will 
7  oxen  weigh  ? 

62.  How  many  bricks  can  be  carried  in  6  loads,  if  1024 
bricks  are  hauled  at  a  load  ? 

63.  A  lot  cost  $  575.     How  much  will  7  lots  cost  at  the 
same  rate  ? 

64.  A  family  of  3   persons   boarded  in  Albany  for  3 
weeks  at  the  rate  of  $  4.75  a  week  for  each  person.     What 
was  the  cost  of  their  board  for  the  whole  time  ? 


60  MULTIPLICATION. 

ORAL  EXERCISES. 
70.    1.    How  many  are  10  times  5  ?     5  times  5  ? 

2.  Since  10  times  5  are  50,  and  5  times  5  are  25,  how 
many  are  (10  +  5),  or  15  times  5  ? 

3.  How  many  are  10  times  8  ?  2  times  8  ?  12  times  8  ? 

4.  How  many  are  10  times  6  ?  4  times  6  ?  14  times  6  ? 

5.  How  many  are  10  times  7  ?  2  times  7?  12  times  7?' 

6.  How  many  are  12  times  4  ?  12  times  5  ?  12  times  3  ? 

7.  How  may  a  number  be  most  conveniently  multiplied 
by  15? 

8.  How  many  are  10  times  9  plus  2  times  9,  or  12 
times  9  ? 

9.  A  foot  contains  12  inches.     How  many  inches  are 
there  in  12  feet  ? 

10.  A  pound  of  sugar  weighs  16  ounces.     How  many 
ounces  do  12  pounds  weigh  ? 

11.  What  will  12  dozen  eggs  cost  at  17  cents  per  dozen  ? 

12.  The  fare  upon  the  railroad  from  a  certain  city  to 
another  is  $  18.     How  much  will  12  tickets  cost  ? 

13.  What  will  18  tons  of  hay  cost  at  $  11  per  ton  ? 

14.  What  will  12  pounds  of  beefsteak  cost  at  18  cents  a 
pound  ? 

15.  How  much  will  13  brooms  cost  at  22  cents  each  ? 

16.  A  farmer  who  owned  15  cows  owned  15  times  as 
many  sheep.     How  many  sheep  did  he  own  ? 

17.  A  man  earned  $18  per  week  for  16  weeks.     How 
much  did  he  earn  in  all  ? 

18.  A  lady  purchased  15  yards  of  cloth,  paying  18  cents 
per  yard  for  it.     How  much  did  she  pay  for  the  whole  ? 


WRITTEN  EXERCISES.  61 

19.  How  many  sheets  of  paper  are  there  in  15  quires, 
since  each  quire  contains  24  sheets  ? 

20.  What  time  will  be  required  to  walk  13  miles,  if  it 
requires  15  minutes  to  walk  one  mile  ? 

21.  A  man  deposited  $15  per  month  in  a  savings  bank. 
How  much  did  his  deposits  amount  to  in  a  year  ? 

22.  A  circular  railroad  is  18  miles  in  length.     How  far 
does  a  train  run  which  makes  the  circuit  15  times  in  a  day  ? 

23.  A  stair  has  15  steps.     How  many  steps  does  a  person 
take  who  goes  up  and  down  it  8  times  per  day  ? 

24.  How  many  are  10  times  5  ?    10  times  6  ?    10  times  7  ? 

25.  What  is  annexed  to  a  number  when  it  is  multiplied 
by  10? 

26.  How  many  are  100  times  2?     100  times  3?     100 
times  4  ? 

27.  What  is  annexed  to  a  number  when  it  is  multiplied 
by  100? 

28.  How  many  are  1000  times  2,  or  2  times  1000  ?     1000 
times  5,  or  5  times  1000  ?     1000  times  9,  or  9  times  1000  ? 

29.  How  is  a  number  multiplied  by  1000  ? 

30.  How  is   a  number    multiplied   by   1   with   ciphers 
annexed  ? 

71.  PRINCIPLE.  —  A  number  is  multiplied  by  10,  100, 1000, 
etc.,  by  annexing  to  the  multiplicand  as  many  ciphers  as  there 
are  in  the  multiplier. 

WRITTEN   EXERCISES. 

72.  What  are  the  products  of  the  following  ? 

1.  274  x  10  381  x  100  9314  x  1000. 

2.  386  x  10  610  x  100  8167  x  1000. 


62  MULTIPLICATION. 

What  are  the  products  of  the  following  ? 

3.  456x10  903x100  7830x10000. 

4.  375x10  857x100  5169x10000. 

5.  319  x  10  310  x  100  6008  x  100000. 

6.  402  x  10  416  x  100  6785  x  100000. 

7.  What  is  the  product  of  3486  x  6000? 

3486  EXPLANATION.  —  Since  6000  is  equal  to  1000  times  6, 

6000     3486  is  first  multiplied  by  6,  giving  a  product  of  20916, 
and  then  that  result  is  multiplied  by  1000  by  annexing 


20916000     three  ciphers. 

8.  393x20  814x300  8345x2000. 

9.  491  x  30  739  x  200  3046  x  8000. 

10.  568  x  20  816  x  700  5104  x  4000. 

11.  715  x  50  793  x  500  6838  x  3000. 

12.  423x70  982x800  7008x8000. 

13.  316  x  40  318  x  300  5913  x  9000. 

14.  629  x  60  427  x  600  8674  x  7000. 

15.  518  x  30  563  x  900  8138  x  6000. 

16.  Multiply  264  by  142. 

EXPLANATION.  —  For  convenience,  the  multiplier  is  written  under 
the  multiplicand,  units  under  units,  tens  under  tens, 


Since  we  cannot  multiply  by  142  at  one  operation, 
142  we  multiply  by  the  parts,  and  then  add  the  products. 

7TT  The  parts  by  which  we  multiply  are  2,  4  tens,  and  1 

hundred. 

10560  2  times  264  equal  528,  the  first  partial  product  ; 

26400  4  tens>  or  40»  times  264  e°.ual  10560,   the  second 

partial  product  ;  1  hundred,  or  100,  times  264  equal 

37488  26400,  the  third  partial  product,   and  the  sum  of 

these,  37488,  is  the  entire  product. 


WRITTEN  EXERCISES.  63 

EXPLANATION.  —  The  ciphers  at  the  right  of  the  partial  products 
may  be  omitted,  the  significant  figures  occupying 
264  their  proper  places  without  the  ciphers. 

-JL42  Thus,  in  multiplying  by  4  tens,  the  lowest  order 

in  the  product  is  tens  ;  hence  the  first  figure  of  the 

528  product  is  written  under  tens,  and  the  rest  in  their 

1056  proper  order. 

In  multiplying  by  hundreds,  the  lowest  order  of 
units  in  the  product  is  hundreds ;  hence  the  first 
37488  figure  of  the  product  is  written  under  hundreds,  and 

the  rest  in  their  proper  order. 

RULE.  —  Write  the  multiplier  under  the  multiplicand,  with 
units  under  units,  tens  under  tens,  etc. 

Multiply  each  figure  of  the  multiplicand  by  each  significant 
figure  of  the  multiplier  successively,  beginning  with  units. 
Place  the  right-hand  figure  of  each  product  under  the  figure 
of  the  multiplier  used  to  obtain  it,  and  add  the  partial  products. 

PROOF.  —  Review  the  work  or  multiply  the  multiplier  by 
the  multiplicand.  If  the  results  agree,  the  work  is  probably 
correct. 

When  there  is  a  cipher  in  the  multiplier,  multiply  by  the  significant 
figures  only,  taking  care  to  place  the  first  figure  in  the  product  under 
the  figure  of  the  multiplier  used  to  obtain  it. 

17. 

3125 

403 

9375         32795       31560 
12500        14055         15780 


1259375  14087795       18936000 

Multiply : 

20.  3842  by  23.  22.    3256  by  41. 

21.  4168  by  32.  23.   4187  by  36. 


64 


MULTIPLICATION. 


Multiply : 

24.  5493  by  27. 

25.  8169  by  46. 

26.  5768  by  35. 

27.  3985  by  71. 

28.  4873  by  64. 

29.  3567  by  58. 

30.  2981  by  49. 

31.  6324  by  37. 

32.  5237  by  65. 

33.  6415  by  79. 

34.  7346  by  84. 

35.  8125  by  34. 

36.  7189  by  67. 

37.  3426  by  73. 

38.  5763  by  85. 

39.  6256  by  99. 

40.  7126  by  83. 

41.  34567  by  325. 

42.  21894  by  216. 

43.  46854  by  406. 

44.  61854  by  314. 

45.  58163  by  248. 

46.  48348  by  232. 

47.  58194  by  397. 

48.  67853  by  206. 

49.  38492  by  387. 

50.  28637  by  341. 

51.  41856  by  416. 


52.  32685  by  507. 

53.  56736  by  693. 

54.  61004  by  372, 

55.  52390  by  4225. 

56.  43257  by  7056. 

57.  654324  by  3097. 

58.  683725  by  5618. 

59.  398647  by  3009. 

60.  318043  by  5960. 

61.  346857  by  6807. 

62.  $468.35  by  3915. 

63.  $  314.27  by  3707. 

64.  $416.32  by  6850, 

65.  $  527.46  by  4837. 

66.  $  304.25  by  5039. 

67.  $  513.64  by  8106. 

68.  $425.38  by  9007. 

69.  $317.24  by  8305. 

70.  $  819.30  by  6804. 

71.  $  364.25  by  5918. 

72.  $371.58  by  6805. 

73.  $296.13  by  6432. 

74.  $324.42  by  3806. 

75.  $418.56  by  6817. 

76.  $524.36  by  4026. 

77.  $327.45  by  3830. 

78.  $416.32  by  4069. 

79.  $487.56  by  5830. 


WRITTEN  EXERCISES.  65 

80.  B  sold  17  firkins  of  butter,  each  firkin  containing  56 
pounds,  at  $  .38  a  pound.     How  much  did  he  receive  for  it  ? 

81.  The  earth  moves  in  its  orbit  19  miles  in  a  second ;  how 
far  does  it  move  in  60  seconds,  or  1  minute  ?     How  far  in 
60  minutes,  or  1  hour  ? 

82.  Two  steamers  start  from  the  same  place  and  sail  in 
opposite  directions,  one  at  the  rate  of  18  miles  an  hour,  and 
the  other  at  the  rate  of  15  miles  an  hour.     How  far  will 
they  be  apart  in  39  hours  ? 

83.  Two   ships  are   7483   miles    apart,   and   are   sailing 
towards  each  other,  one  at  the  rate  of  46,  the  other  at  the 
rate  of  53  miles  a  day.     How  far  will  they  be  apart  at 
the  end  of  73  days  ? 

84.  I  bought  156  barrels  of  flour  for  $  1015.     Finding 
32  barrels  of  it  worthless,  I  sold  the  remainder  at  $9  a 
barrel.     Did  I  gain  or  lose,  and  how  much  ? 

85.  In  an  orchard  there  are  117  rows  of  trees,  and  each 
row  contains  69  trees.     How  many  trees  are  there  in  the 
orchard  ? 

86.  A  freight  train  consists  of  26  cars ;  each  car  contains 
82  barrels  of  flour,  and  each  barrel  of  flour  weighs  196 
pounds.     How  many  pounds  of  flour  in  the  entire  cargo  ? 

87.  Mr.  Burns  earns  $  37.60  a  week.     His  expenses  are 
$  19.50  a  week.     How  much  can  he  save  in  26  weeks  ? 

88.  It  requires  1972  pickets  to  fence  one  side  of  a  square 
lot.     How  many  pickets  will  be  required  to  fence  17  lots  of 
the  same  size  and  shape  ? 

89.  I  sold  from  my  farm  270  bushels  of  wheat  at  $  .98 
per  bushel,  300  bushels  of  corn  at  $.55  per  bushel,  175 
bushels  of  oats  at  $.35  per   bushel,  and   60   bushels   of 
potatoes  at  $.45  per  bushel.     What  was  the  value  of  the 
crops  sold  from  the  farm  ? 

STAND.   AR.  —  5 


66  MULTIPLICATION. 

90.  A  lady  bought  at  a  store  12  yards  of  silk  at  87  cents 
a  yard,  36  yards  of  ribbon  at  15  cents  a  yard,  and  17  yards 
of  muslin  at   8   cents   a  yard.     She  gave  the  clerk  a  20- 
dollar  bill  to  pay  for  it.     How  much  change  should  she 
receive  ? 

91.  A  has  $65,  B  has  three  times  as  much  as  A,  and  C 
has  as  much  as  both,  lacking  $  25.   How  much  have  they  all  ? 

92.  I  bought  80  tons  of  coal  at  $3.25  a  ton,  and  paid 
freight  amounting  to  $  13.75.     I  sold  55  tons  for  $  3.80  a 
ton,  and  the  balance  for  $3.95  a  ton.     How  much  did  I 
make  on  the  coal  ? 

93.  Mr.  Hughes  sold  a  farm  of  160  acres  at  $  97.50  per 
acre,  and  received  in  payment  a  note  for  $  3765.27,  and  the 
rest  in  cash.     How  much  cash  did  he  receive  ? 

94.  A  dealer  bought  700  bushels  of  wheat  at  $.95  a 
bushel.     He  sold  256  bushels  at  $  1.10  a  bushel,  44  bushels 
at  $1.15  a  bushel,  and  the  remainder  for   $394.      How 
much  did  he  gain  on  the  entire  sale  ? 

95.  A  speculator  bought  640  acres  of  land  at  $20  an 
acre.     He  sold  at  one  time  350  acres  at  $25  an  acre;  at 
another  time  150  acres  at  $28  an  acre.     What  must  he 
receive  for  the  remainder  to  gain  $  4500  on  the  purchase  ? 

96.  A  farmer  who  raised  1160  bushels  of  oats  kept  75 
bushels  for  seed,  and  enough  to  winter  12  horses,  allowing 
45  bushels  to  each  horse,  and  sold  the  remainder.     How 
many  bushels  did  he  sell  ? 

97.  A  cloth  merchant  sold  three  lots  of  cassimeres,  the 
first  containing  19  pieces  of  28  yards  each,  at  $1.75  per 
yard ;  the  second  containing  14  pieces  of  27  yards  each,  at 
$  1.87  per  yard ;  and  the  third  containing  40  pieces  averag- 
ing 25  yards  each,  at  $  1.95  per  yard.     What  was  the  value 
of  the  whole  ? 


DIVISION. 


73.  1.   How   many   groups   of   2   pencils    each  can    be 
formed  from  6  pencils  ?     How  many  2's  are  there  in  6  ? 

2.  How  many  groups  containing  3  marbles  each  can  be 
formed  from  9  marbles  ?     How  many  3's  are  there  in  9  ? 

3.  How  many  3's  are  there  in  6  ?     How  many  4's  in  8  ? 
How  many  times  is  3  contained  in  6  ?     2  in  6  ?     4  in  8  ? 

4.  How  many  cents  will  each  child  have  when  12  cents 
are  divided  equally  among  4  children  ?      How  many  4's 
in  12? 

5.  If  16  cents  are  divided  equally  among  8  children, 
how  much  will  each  have  ? 

6.  What  have  you  been  doing  with  the  above  numbers  ? 

7.  How  many  are  five  6's  ?     How  many  6's  in  30  ? 

8.  How  many  are  seven  4's  ?     How  many  4's  in  28  ? 

9.  How  many  are  six  3's  and  2  ?    How  many  3's  in  20  ? 
10.   How  many  5's  are  there  in  22?     How  many  are 

four  5's  and  2  ? 

74.  The  process  of  finding  how  many  times  one  number 
is  contained  in  another  is  called  Division ;  or, 

Division  is  the  process  of  separating  a  number  into  equal 
parts.     This  process  is  usually  called  Partition. 

75.  The  number  to  be  divided  is  called  the  Dividend. 

76.  The  number  by  which  we  divide  is  called  the  Divisor. 

67 


68  DIVISION. 

77.  The  result  obtained  by  division  is  called  the  Quotient. 
It  shows  how  many  times  the  divisor  is  contained  in  the 

dividend. 

78.  The  part  of  the  dividend  remaining  when  the  division 
is  not  exact  is  called  the  Remainder. 

79.  The  Sign  of  Division  is   — .     It  is  read  divided  by. 
When  it  is  placed  between  two  numbers,  it  shows  that 
the  one  on  the  left  is  to  be  divided  by  the  .one  on  the  right. 

Thus,  63  -H  9  is  read  63  divided  by  9. 

Division  is  also  indicated  by  writing  the  dividend  above 
the  divisor. 

Thus,  6¥3-  indicates  that  63  is  to  be  divided  by  9. 

80.  PRINCIPLES.  —  1.   When  the  dividend  and  divisor  are 
like  numbers,  the  quotient  must  be  an  abstract  number. 

2.  The  dividend  and  remainder  are  like  numbers. 

3.  The  product  of  the  divisor  and  quotient,  plus  the  re- 
mainder, is  equal  to  the  dividend. 

An  example  like  "  How  many  6's  are  there  in  30  ?  "  may  be  solved 
by  subtraction,  but  it  requires  a  longer  time  than  by  division.  Hence, 
division  may  be  regarded  as  a  short  method  of  subtracting  equal  numbers. 

The  same  example  may  be  solved  readily  by  recalling  the  products 
in  multiplication.  When  we  wish  to  know  how  many  sixes  there  are 
in  30,  if  we  recall  the  fact  that  five  sixes  are  30,  the  answer  is  found 
at  once.  Hence,  division  is  the  converse  of  multiplication. 

DRILL    EXERCISES. 

81.  Tell  the  quotients  of  the  following  instantly  : 


40-   4. 

15-*-    5. 

33-   3. 

36-:-      6. 

42-*-    7. 

25-*-   5. 

30  -*-  10. 

32-   4. 

77-*-    7. 

60-10. 

30-*-   6. 

24-*-   2. 

40  -*-  10. 

99  -*-  11. 

56-*-    8. 

18-*-   9. 

72-*-    6. 

64-*-    8. 

28-*-    7. 

36-*-   3. 

64-+-   6. 

36  -r-  12. 

65-*-   5. 

72-    9. 

14-*-   2. 

ORAL  EXERCISES.  69 


20  -  10. 

21-  7. 

48  -  12. 

50  -  10. 

22-  2. 

8-4. 

48—  4. 

24-  4. 

27-j-  9. 

32-  8. 

20-  2. 

42-  6. 

16  -r-  2. 

66-  6. 

60-5. 

96-  8. 

44  -  11. 

24-  6. 

60  -  12. 

88  -  11. 

6-  3. 

28  -r-  4. 

30-  5. 

63-  7. 

40-  5. 

22  -  11. 

36  -r-   4. 

66  -  11. 

48-  8. 

44-  4. 

12-  2. 

70-10. 

8-2. 

49-  7. 

96  -  12. 

20-  4. 

63-  9. 

81  -r-  9. 

35-  7. 

24-  3. 

24  -  12. 

88-  8. 

35-  5. 

90  —  10. 

45-  9. 

18-  3. 

55  -  11. 

80  -  10. 

48-  6. 

108-  9. 

60-  6. 

84-  7. 

10-  2. 

110  -  11. 

121  -  11. 

15-  3. 

16-  4. 

77-11. 

80-  8. 

108  -  12. 

56-  7. 

14-  7. 

6-2. 

99-  9. 

21-  3. 

90-  9. 

4-  2. 

36  -f-  9. 

40-  8. 

132  -  12. 

70-  7. 

16-  8. 

84-12. 

50-  5. 

120  -  10. 

72-  8. 

72  -  12. 

18-  2. 

27-  3. 

110  -  10. 

20-  5. 

30-  3. 

54-  9. 

100  +-  10. 

120  -  12. 

12-  6. 

18-  6. 

12-  3. 

24-  8. 

132  -  11. 

33  -f-  11. 

10  -f-  5. 

12-  4. 

9-  3. 

144-12. 

ORAL   EXERCISES. 

82.   1.    At  5   cents   apiece,   how   many  pencils   can   be 
bought  for  25  cents  ?     For  35  cents  ?     For  45  cents  ? 

2.  How  many  yards  of  cloth,  at  8  cents  per  yard,  can  be 
bought  for  40  cents  ?     How  many  8's  are  there  in  40  ? 

3.  A  field  of  70  acres  was  divided  into  lots  containing 
7  acres  each.     How  many  lots  were  there  ? 

4.  A   dollar   was   changed   into   10-cent   pieces.      How 
many  were  there  ?     How  many  7's  are  there  in  56  ? 

5.  A  girl's  entire  expenses  at  boarding  school  were  $6 
per  week.     For  how  many  weeks  will  $  48  pay  ? 

6.  How  many  5-cent  pieces  are  there  in  a  dime  ? 


70  DIVISION. 

7.  What  is  paid  per  yard  for  cloth  if  I  get  7  yards  for 
63  cents  ?     How  many  7's  in  63  ?     8's  in  72  ? 

8.  A  girl  visited  at  her  uncle's  for  42  days.     How  many 
weeks  was  that  ?     How  many  7's  in  42  ?     6's  in  42  ? 

9.  A  freight  train  averaged  9  miles  per  hour.     In  how 
many  hours  would  it  run  72  miles  ?     How  many  8's  in  72  ? 

10.  An  orchard  contained  54  trees,  arranged  in  9  rows. 
How  many  trees  were  there  in  each  row  ? 

11.  When  oranges  sell  at  8  cents  per  dozen,  how  many 
dozen  can  be  bought  for  80  cents  ? 

12.  A  grocer's  profit  upon  a  pound  of  tea  was  7  cents. 
How  many  pounds  must  he  sell  to  gain  63  cents  ? 

13.  A  stage  coach  went  8  miles  per  hour.     How  long 
would  it  require  to  go  80  miles  ?     How  many  5's  in  50  ? 

14.  A  school-room  contained  54  seats,  arranged  in  6  rows. 
How  many  seats  were  there  in  each  row  ? 

15.  When  flour  sells  at  $  6  per  barrel,  how  many  barrels 
can  be  purchased  for  $  48  ? 

16.  How  many  engravings,  worth  $  7  each,  can  be  bought 
for  $  49? 

17.  When  coal  sells  at  $5  per  ton,  how  many  tons  can  I 
buy  for  $  50  ? 

18.  How  many  rods  of  fence,  at  $  8  per  rod,  can  be  made 
for  $64? 

19.  A  man's  wages   are   $9  per  week.     In   how  many 
weeks  can  he  earn  $  45  ? 

20.  If  5  pounds  of  sugar  cost  25  cents,  how  many  pounds 
can  be  bought  for  40  cents  ? 

21.  My  wood  cost  me  $  40,  at  the  rate  of  $5  per  cord. 
How  many  cords  did  I  purchase  ? 


ORAL  EXERCISES.  71 

22.  If  writing  books  sell  at  8  cents  each,  how  many  can 
be  bought  for  72  cents  ? 

23.  When  5  yards  of  cloth  cost  35  cents,  how  many 
yards  can  be  purchased  for  63  cents  ? 

24.  If  4  quarts  of  milk  sell  for  24  cents,  how  many  quarts 
can  you  buy  for  42  cents  ? 

25.  A  boy  paid  21  cents  for  3  quarts  of  chestnuts.     How 
many  quarts  could  he  have  purchased  for  35  cents  ? 

26.  At  the  rate  of  6  pencils  for  48  cents,  how  many 
pencils  could  be  bought  for  32  cents  ? 

27.  A  man  gave  54  cents  to  some  beggars,  giving  9  cents 
to  each.     How  many  beggars  were  there  ? 

28.  If  6  packages  of  paper  cost  42  cents,  how  many  such 
packages  can  be  obtained  for  56  cents  ? 

29.  If  8  pounds  of  lard  cost  48  cents,  how  many  pounds 
Can  be  bought  for  42  cents  ? 

30.  How  many  sheep  can  be  bought  for  $  35,  if  6  sheep 
cost  $30? 

31.  If  7  tables  can  be  bought  for  $  56,  how  many  tables, 
at  the  same  rate,  can  be  purchased  for  $  72  ? 

32.  I  bought  7  calves  for  $  35.     How  much  was  that  per 
head? 

33.  Mary  had  9  packages  of  paper  of  the  same  number 
of  sheets,  and  in  all  she  had  54  sheets.     How  many  sheets 
were  there  in  each  package  ? 

34.  How  many  sheep,  at  $  4  per  head,  can  be  purchased 
by  a  man  who  has  $  42  ?  Ans.  10  sheep,  and  $  2  left. 

35.  A  man  who  worked  for  $3  per  day   earned   in   a 
certain  time  $  32.     How  many  days  did  he  work  ? 

Ans.  lOf  days. 


72  DIVISION. 

83.  From  problems  34  and  35  it  is  apparent  that  the 
remainder  should  sometimes  be  written  after  the  quotient, 
and  sometimes  as  a  part  of  it. 

When  the  remainder  is  written  as  part  of  the  quotient, 
it  is  expressed  by  placing  the  divisor  under  it  with  a  line 
between  them,  as  in  the  answer  to  problem  35. 

84.  When  anything  is  divided  into  two  equal  parts,  each 
part  is  called  one  half ;  when  into  three  equal  parts,  one 
third;  when  into  four  equal  parts,  one  fourth,  etc. 

85.  One  or  more  of  the  equal  parts  of  anything  is  called 
a  Fraction. 

Fractions  are  expressed  as  follows : 

One  half  by  £.          One  third  by  J.         Two  thirds  by  f . 
One  fourth  by  J.       Three  fourths  by  f .  Two  fifths  by  f . 
Five  eighths  by  -|.    Seven  ninths  by  -J.    Five  elevenths  by  y5T. 

86.  1 .    E/ead  the  following  fractional  expressions : 

A      T'T      «      «      H      «      it      H 
«      H      if      «      «      tt      tt      tt 

3TF  "6T  "2T  T9  "ST  TS  9T  Tl5 

2.  What  is  J  of  10?  12?  16?  18?  20? 

3.  What  is  £  of  15  ?  18?  24?  27?  30? 

4.  What  is  J  of  12?  16?  20?  36?  40? 

5.  What  is  |  of  12  ?  18?  24?  36?  42? 

6.  What  is  £  of  16  ?  24?  40?  64?  72? 

7.  What  is  £  of  10?  30?  35?  45?  50? 

8.  Whatis|of21?  35?  42?  56?  49? 


ORAL   EXERCISES.  73 

9.   A  man  earned  $  45  in  9  days.     What  did  lie  receive 
per  day  ? 

10.  In  7  weeks  there  are  49  days.     How  many  are  there 
in  one  week  ? 

11.  How  much  does  each  door  cost  me,  if  I  pay  $56  for 
8  of  them  ? 

12.  If  5  barrels  of  flour  are  worth  $  30,  what  is  the  price 
per  barrel  ? 

13.  If  6  Christmas  cards  cost  48  cents,  what  will  be  the 
cost  of  one  at  the  same  rate  ? 

14.  A  lady  paid  64  cents  for  8  yards  of  muslin.     How 
much  did  she  pay  per  yard  ? 

15.  A  man's  wages  for  6  days'  work  amounted  to  $  24. 
How  much  did  he  receive  per  day  ? 

16.  Mr.  B  paid  $  35  for  7  weeks'  board.     How  much  did 
the  boarding  cost  him  per  week  ? 

17.  If  a  farm  hand  should  earn  $  20  for  laboring  4  weeks, 
how  much  would  he  earn  in  a  week  ? 

18.  In  7  weeks  a  man  saved  $  49.     If  he  saved  the  same 
amount  each  week,  what  did  he  save  per  week  ? 

19.  If  8  barrels  of  sugar  are  worth  $  64,  what  is  the 
price  per  barrel  ? 

20.  If  9  writing  desks  cost  $  54,  what  is  the  cost  of  one 
writing  desk  ? 

21.  A   merchant   paid  $36  for  9  kegs  of  nails.     How 
much  did  he  pay  per  keg  ? 

22.  If  9  shawls  of  a  certain  quality  cost  $  72,  what  must 
you  pay  for  1  shawl  of  the  same  quality  ? 


74 


DIVISION. 


Divisor.   Dividend. 

4)1384 
Quotient.    346 


WRITTEN   EXERCISES. 

87.  1.   Divide  1384  by  4. 

EXPLANATION.  — For  convenience,  the  divisor  is  written  at  the  left 
of  the  dividend,  with  a  line  between  them,  and  the  quotient  either 

under  the  dividend  or  at  the  right  of  it,  with  a  line 

between  them. 

4  is  not  contained  in  1  thousand  any  thousand 

times,  therefore  the  quotient  cannot  contain  units 

of  any  order  higher  than  hundreds.  Hence  we  find 
how  many  times  4  is  contained  in  all  the  hundreds  of  the  dividend. 
1  thousand  plus  3  hundreds  are  13  hundreds.  4  is  contained  in  13 
hundreds  3  hundreds  times,  with  a  remainder  of  1  hundred.  We  write 
the  3  hundreds  of  the  quotient  under  the  hundreds  of  the  dividend. 

1  hundred,  the  remainder,  plus  8  tens  are  18  tens.     4  is  contained 
in  18  tens  4  tens  times,  with  a  remainder  of  2  tens.    We  write  the  4  tens 
of  the  quotient  under  the  tens  of  the  dividend. 

2  tens,  the  remainder,  plus  4  units  are  24  units.     4  is  contained 
in  24  units  6  units  times,  and  the  6  is  placed  under  the  units  of  the 
dividend. 

Hence  the  quotient  is  346. 

PROOF.  —  The  quotient,  346,  multiplied  by  4,  the  divisor,  is  1384. 
Hence  the  work  is  correct  (Prin.  3). 

88.  When  examples  in  division  are  solved  without  writ- 
ing the  products  or  remainders,  the  process  is  called  Short 
Division. 


Divide  by  short  division : 


3. 

6)216276 
36046 


4. 

5)31250 
6250 


6.  3624-4. 

7.  2135-5. 

8.  3852-3. 

9.  2835-5-5. 


10.  3258  —  6. 

11.  4264-4. 

12.  3042-6. 

13.  6105  —  5. 


5. 

8)403287 
50410|- 

14.  4506-3. 

15.  4864-4. 

16.  5274-6. 

17.  5325—5. 


WRITTEN  EXERCISES.  75 

Divide  by  short  division : 

18.  7938-5-3.  37.   ±J£*.  56.      619381-5-9. 

19.  4824-5-8.  38.   41M.  57.      492347-4-7. 

20.  3160-8-4.  39.   &if±  58.      583629-5-6. 

21.  5830-4-5.  40.    A^i.  59.      510432-5-8. 

22.  6876-5-3.  41.   ±j£i.  60.      617386-5-9. 

23.  8676-5-6.  42.    ^p.  61.    $432.15-5-5. 

24.  5832-5-8.  43.    ^i.  62.    $346.05-5-9. 

25.  4735-5-5.  44.    Jup.  63. '  $  514.29  -4-  7. 

26.  5872-5-4.  45.   up*.  64.    #683.24-*- 8. 

27.  3920-*- 8.  46.    iflp.  65.    $  516.89  -f-  6. 

28.  4163-1-3.  47.   £-4/±  66.    $  342.85 -j- 4. 

29.  3618-4-6.  48.    **£&.  67.    $397.54-*- 9. 

30.  2944 --4.  49.    Mp.  68.    $819.32^-7. 

31.  5112^-3.  50.   i^UL.  69.    $509.08-5-6. 

32.  3972^-6.  51.   ^^.  70.    $314.05-5-9. 

33.  5984-5-4.  52.   MM.  71.    $429.65-5-8. 

34.  4626-5-6.  53.    Mp.  72.    $734.74-5-6. 

35.  7255-4-5.  54.    46^4.  73.    $529.86-5-7. 

36.  7384-5-8.  55.   uf*.  74.    $393.81-5-9. 

75.  A  man  divided  his  fortune  of  $7616  equally  among 
his  6  children.     What  was  the  share  of  each  ? 

76.  There  are  7  days  in  a  week.  How  many  weeks  are 
there  in  1015  days  ? 

77.  At  $6  a  barrel,  how  many  barrels  of  flour  can  be 
purchased  with  $  1422. 

78.  If  8  horses  of  equal  value  are  worth  $2352,  what  is 
each  horse  worth  ? 


76  DIVISION. 

79.  A  man  has  3248  acres  of  land  lying  in  7  equal  tracts. 
How  many  acres  are  there  in  each  tract  ? 

80.  Mr.  B  bought  6  cows  for  $182.10.     What  was  the 
average  price  of  each  cow  ? 

81.  How  many  tons  of  coal,  at  $5  a  ton,  can  be  bought 
for  $3125? 

82.  I  bought  8  yards  of  broadcloth  for  $41.60.     What 
was  the  price  per  yard  ? 

83.  There  are  8  quarts  in  one  peck.     How  many  pecks 
are  there  in  9136  quarts  ? 

84.  If  6  lots  are  worth  $  7470,  what  is  the  average  value 
of  each  lot  ? 

85.  Divide  an  estate  of  $  15302  equally  among  7  heirs. 

86.  A  lady  bought  at  a  store  a   number   of  yards   of 
muslin  at  8  cents  a  yard,  and   paid  $3.84  for   it.     How 
many  yards  did  she  buy  ? 

87.  If  7  plows  are  worth  $94.50,  what  is  the  average 
price  of  a  plow  ? 

88.  If  6  wagons  of  the  same  make,  quality,  and  size,  are 
worth  $  454.80,  what  is  the  value  of  each  ? 

89.  In  a  nail  factory  73,455  nails  were  made  by  5  boys 
in  2  hours.     What  was  the  average  number  made  by  each 
boy? 

90.  There  are  9  square  feet  in  a  square  yard.      How 
many  square  yards  are  there  in  36,783  square  feet  ? 

ORAL    EXERCISES. 

89.    1.    At  10  cents  each,  how  many  melons  can  be  bought 
for  90  cents  ? 

2.  At  $  10  each,  how  many  coats  can  be  bought  for  $  60  ? 

3.  If  a  book  costs  30  cents,  how  many  can  be  bought 
for  60  cents  ? 


WRITTEN  EXERCISES 


77 


4.  If  a  man  earns  $20  per  week,  in  how  many  weeks 
can  he  earn  $  60  ? 

5.  If  there  are  80  apple   trees  arranged  in  20  rows, 
how  many  trees  are  there  in  each  row  ? 

6.  How  many  days  will  I  require  to  travel  150  miles, 
if  I  travel  50  miles  per  day  ? 

7.  How  many  10's  are  there  in  50  ?     In  60  ?     In  70  ? 
In  80?     In  130?     In  150  ?      In  250  ?     In  350  ?     In  600  ? 
In  800  ? 

8.  Since  10  is  contained  5  times  in  50,  6  times  in  60, 
35  times  in  350,  80  times  in  800,  how  may  a  number  be 
divided  by  10  ? 

9.  How  many  100's  are  there  in  600?     In  800?     In 
3700  ?     In  8500  ? 

10.  Since  100  is  contained  6  times  in  600,  8  times  in 
800,  37  times  in  3700,  and  85  times  in  8500,  how  may  a  num- 
ber be  divided  by  100  ? 

90.  PRINCIPLE.  —  A  number  may  be  divided  by  10,  100, 
1000,  etc.,  by  cutting  off  from  the  right  of  the  dividend  as 
many  figures  as  there  are  ciphers  upon  the  right  of  the  divisor. 


91.   Divide: 
1. 

1|00)275|00 
275 

4.  46853-100. 

5.  39278-100. 

6.  38546-100. 

7.  46850-100. 

8.  31700-100. 

9.  68543-5-1000. 


WRITTEN    EXERCISES. 

2. 

1|00)394|63 
394,% 

10.  31927  -r-  1000. 

11.  41687-1000. 

12.  38125  -5- 1000. 

13.  41736  -T- 1000. 

14.  54286-1-1000. 

15.  31854-10000. 


16.  48653  - 10000. 

17.  31925-10000. 

18.  46874  -  10000. 

19.  72840  - 10000. 

20.  61735  - 10000. 

21.  42856-10000. 


78  DIVISION. 

22.    Divide  72595  by  400. 

EXPLANATION.  —  After  cutting  off  the  two  ciphers  from  the  right 

of  the  divisor  and  two  figures  from  the  right  of  the  dividend,  the 

remaining  part  of  the  dividend  is  divided  by  4,  giving 

4|00)725[95       a  quotient  of  181  and  1  hundred  remainder.     1  hun- 

181  195.     dred  plus  the  partial  remainder,  95,  cut  off,  gives  the 

entire  remainder  195. 
Hence  the  quotient  is  181  and  195  remainder  or 


Divide  : 

23.  3845  by  30.  28.  98357  by  400.  33.  316857  by  2000. 

24.  4927  by  40.  29.  81654  by  700.  34.  415684  by  4000. 

25.  6839  by  50.  30.  32718  by  600.  35.  238719  by  6000. 

26.  4168  by  70.  31.  42513  by  900.  36.  418576  by  9000. 

27.  6985  by  90.  32.  24678  by  800.  37.  382456  by  7000. 

38.    Divide  7975  by  26. 

EXPLANATION.  —  26  is  not  contained  in  7  thousands  any    thousands 

times  ;  hence  the  thousands  are  united  with  the  hundreds,  making 

79  hundreds.    26  is  contained  in  79  hun- 

dreds, 3  hundreds  times  with  a  remainder. 

Divisor.  Dividend.  Quotient.  The  3  ^midreds  are  written  in  the  quo- 
9*n  7Q7X  /QHA19  tient'  and  the  divisor  1S  multiplied  by 
^Oj  <y/0  (.oUOyg-  them,  giving  a  product  of  78  hundreds,  or 

78  7  thousands  and  8  hundreds,  which  are 

written  under  units  of  the  same  order  in 

the  dividend.     Subtracting,  there  is  a  re- 

156  mainder  of  1  hundred. 

—  The   1   hundred  is  united  with  the  7 

tens,  making   17   tens.     26  is  not  con- 

tained hi  17  tens  any  tens  times  ;  there- 

fore there  are  no  tens  in  the  quotient,  and  a  cipher  is  written  in  tens' 
place  in  the  quotient. 

The  17  tens  are  united  with  the  5  units,  making  175  units.  26  is 
contained  in  175  units  6  times  with  a  remainder.  The  6  is  written  in 
units'  place  in  the  quotient,  and  the  divisor  multiplied  by  it,  giving  a 
product  of  156  units,  or  1  hundred,  5  tens,  and  6  units,  which  are 
written  under  units  of  the  same  order  in  the  partial  dividend.  Sub- 


WRITTEN  EXERCISES.  79 

tracting,  there  is  a  remainder  of  19.     The  remainder  is  written  over 
the  divisor  as  part  of  the  quotient. 
Hence  the  quotient  is  306£f . 

PROOF.  —306  x  26 +  19 =7975.    Hence  the  work  is  correct  (Prin.  3). 

92.  When  the  steps  in  the  solution  of  an  example  in 
division  are  written,  the  process  is  called  Long  Division. 

RULE.  —  Write  the  divisor  at  the  left  of  the  dividend  with  a 
curved  line  between  them. 

Find  how  many  times  the  divisor  is  contained  in  the  fewest 
figures  on  the  left  hand  of  the  dividend  that  will  contain  it,  and 
write  the  quotient  on  the  right. 

Multiply  the  divisor  by  this  quotient,  and  place  the  product 
under  the  figures  divided.  Subtract  the  result  from  the  partial 
dividend  used,  and  to  the  remainder  annex  the  next  figure  of 
the  dividend. 

Divide  as  before,  until  all  the  figures  of  the  dividend  have 
been  annexed  to  the  remainder. 

If  any  partial  dividend  will  not  contain  the  divisor,  write  a 
cipher  in  the  quotient,  then  annex  the  next  figure  of  the  divi- 
dend, and  proceed  as  before. 

If  there  is  a  remainder  after  the  last  division,  write  it  after 
the  quotient,  or  with  the  divisor  under  it  as  part  of  the  quotient. 

PROOF.  —  Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder,  if  any.  If  the  work  is  correct,  the 
result  will  equal  the  dividend. 

1.  To  find  the  quotient  figure,  see  how  many  times  the  first  figure 
of  the  divisor  is  contained  in  the  first  figures  of  the  partial  dividend  that 
will  contain  it,  making  allowance  for  the  addition  of  the  tens  from 
the  product  of  the  second  figure  of  the  divisor. 

2.  If  the  product  of  the  divisor  by  the  quotient  figure  is  greater 
than  the  partial  dividend  from  which  it  is  to  be  subtracted,  the  quo- 
tient figure  is  too  large. 

3.  Each  remainder  must  be  less  than  the  divisor  ;  otherwise  the 
quotient  figure  is  too  small. 

4.  When  there  is  no  remainder,  the  divisor  is  said  to  be  exact. 


80 


DIVISION. 


Find  the  quotients  of  : 

39.  3443-*- 11.        53.   41676  -5-  92. 

40.  1728-5-12.        54.    54250-5-62. 

41.  4536-5-21.       55.   52808-5-82. 

42.  11904-5-31.       56.   52056-5-72. 

43.  25133-5-41.       57.   52542-5-63. 

44.  12036-5-51.       58.   46216-5-53. 

45.  21045-5-61.       59.   51875-5-83. 

46.  30885-5-71.       60.   42628-5-74. 

47.  26406-5-81.       61.   50560-5-64. 

48.  33852-5-91.       62.   48168-5-54. 

49.  18832-5-22.       63.    65940-5-84. 

50.  24192-5-32.       64.    63168-5-94. 

51.  32004-5-42.       65.   31080-5-35. 

52.  45292  -5-52.       66.   41220 ---45. 

81.  11413383-5-201.  93. 

82.  21346152 -v- 401.  94. 

83.  23143276-5-601.      95. 

84.  30412345-5-510.      96. 

85.  34103528-5-610.      97. 

86.  51234432-5-711.      98. 

87.  45161812-5-802.      99. 

88.  36784533-5-603.     100. 

89.  25416259-5-613.     101. 

90.  31425634-5-520.     102. 

91.  48765432-5-730.     103. 

92.  45346571-5-831.     104. 


67.  106950-5-75. 

68.  108320  H- 85. 

69.  126618-^-94. 

70.  211044-5-86. 

71.  262656-5-76. 

72.  507558-5-87. 

73.  361437-5-57. 

74.  364450-5-37. 

75.  429436-5-49. 

76.  460346-5-58. 

77.  513282-5-66. 

78.  522786-5-89. 

79.  727748-5-98. 

80.  794061-5-83. 
64785278  -5-  911. 
68475384  -5-  932. 
73146254  -5-  807. 
38765893  -*-  717. 
54563524  ---  722. 
61248638  -5-  834. 
75143920  -*-  950. 
43161726  --  856. 
63517429  -5-  923. 
58463471  -*-  731. 
39158167  -5-  473. 
41132516  -5-  566. 


WRITTEN  EXERCISES.  81 

Find  the  quotients  of : 

105.  31846489-1-1047.  109.  241671382-5-8346. 

106.  57169438-5-3109.  110.  364128796-5-9215. 

107.  84365712-5-5186.  111.  403214571-5-7843. 

108.  136184527-5-7408.  112.    754326840-5-9618. 

113.  If  312  cows  are  worth  $11232,  what  is  the  value 
of  each  cow  ? 

114.  A  ship  sails  7812  miles  in  36  days.     How  far  does 
it  sail  in  1  day  ? 

115.  If  49  horses  are  worth  $  16,758,  what  is  the  average 
value  of  each  ? 

116.  Into  how  many  farms  of  144  acres  each  can  a  tract 
of  land  containing  10,368  acres  be  divided  ? 

117.  At  $82  per  acre,  how  many  acres  of  land  can  be 
bought  for  $  317,094  ? 

118.  How  many  colts,  at  $95  each,  can  be  bought  for 
$42,750? 

119.  There  are  128  cubic  feet  in  one  cord  of  wood.    How 
many  cords  of  wood  are  there  in  69,248  cubic  feet  ? 

120.  The  president  of  the  United  States  receives  a  yearly 
salary  of  $50,000.     How  much  is  that  a  day  ? 

121.  A  man  traveled  83,280  rods.     How  many  miles  did 
he  travel,  there  being  320  rods  in  a  mile  ? 

122.  If  a  field  of  109  acres  produces  3379   bushels   of 
wheat,  what  is  the  average  yield  per  acre  ? 

123.  A  farmer  desires  to  exchange  150  acres  of  land,  at 
$  117  an  acre,  for  woodland,  at  $  39  per  acre.     How  many 
acres  will  he  get  ? 

STAND.    AR.  6 


82  DIVISION. 

124.  The  product  of  two  numbers  is  290,625.     One  of 
the  numbers  is  465 ;  what  is  the  other  ? 

125.  Mr.  King  has  $8000  ;  he  buys  a  house  for  $4500, 
and  some  land  at  $  140  an  acre.     How  many  acres  of  land 
can  he  buy  ? 

126.  The   circumference  of  the  earth  is  25,000  miles; 
how  long  would  it  take  to  travel  round  it,  going  at  the  rate 
of  125  miles  per  day  ? 

127.  If  the  circumference  of  a  wagon  wheel  is  15  feet, 
how  many  turns  will  the  wheel  make  in  going  52800  feet, 
or  ten  miles  ? 

128.  A  man  gave  $5760  and  128  cows,  worth  $50  each, 
for  land  valued  at  $  64  per  acre.     How  many  acres  did  he 
receive  ? 

129.  A  man  wills  $7000  to  his  wife,  $2000  to  a  church, 
$  1000  to  a  school,  and  the  remainder  to  his  8  children,  in 
equal  shares.     What  does  each  child  receive,  the  fortune 
being  $42,720? 

130.  The  area  of  the  state  of  North  Carolina  is  52,250 
square  miles,  and  the  population,  according  to  the  census 
of  1890,  was  1,617,947.     About  how  many  persons,  on  an 
average,  were  there  living  on  a  square  mile  ? 

131.  The  state  of  New  York  has  an  area  of  49,170  square 
miles;    Rhode  Island  has  an  area  of  1250  square  miles. 
Into  how  many  states  of  the  size  of  Ehode  Island  could  New 
York  be  divided,  and  how  many  square  miles  would  be  left 
over? 

132.  Texas  has  an  area  of  265,780  square  miles.     Into 
how  many  states  of  the  size  of  New  York  could  Texas  be 
divided,  and  how  many  square  miles  would  be  left  over  ? 


WRITTEN  EXERCISES.  83 

93.  When  several  numbers  are  to  be  treated  as  one  num- 
ber, they  are  included  in  parentheses  (  ),  in  braces  |j,  in 
brackets  [  J,  or  placed  under  the  vinculum  . 

These  signs  are  called  Signs  of  Aggregation. 

The  expressions  included  by  these  signs  are  to  be  treated  as 
a  single  number. 

Thus,  (5  +  7)x5or5  +  7x5  means  that  the  sum  of  5  and  7  is 
to  be  multiplied  by  5. 

The  parts  of  an  expression  connected  by  the  signs  +  or 
—  are  the  Terms  of  the  expression. 

Thus,  the  expression  3+6  contains  two  terms.  (3 + 6)  x  3  -  (3 + 2)  x  2 
also  contains  but  two  terms,  for  the  numbers  in  the  parentheses  are  each 
treated  as  a  single  term. 

1.  5  +  10-5=5+2  or  7;  but  (5 +10) -5= 15-5  or  3. 

2.  2+3x10-4=2+30-4  or  28;   but  (2+3)xlO-4= 

50-4  or  46  and  (2+3)  x  (10-4)  =  5x6  or  30. 

Hence,  to  find  the  value  of  such  expressions : 

1st.  Simplify  the  expressions  within  the  parentheses  by  per- 
forming the  operations  indicated.  2d.  Further  simplify  each 
term,  if  necessary,  by  performing  the  multiplication  or  division 
indicated.  3d.  Combine  the  terms. 

When  one  parenthesis  includes  another,  remove  the  inner  one  first. 

Find  the  value  of : 

3.  (3  +  4)x8.  8.    (3  +  4)  x  5 -(5  + 4)  — 3. 

4.  (3  +  4)x(8-5).  9.   (6  +  8)-2  +  3x5  +  4. 

5.  (5  +  7)_i_(4_2).         10.  7x4  +  3- (12 -4)^2. 

6.  (8 +  3) -(6 -2).         11.  2  + 12-4- (7 +  8-4) -3. 

7.  (9-3)-(7-5).         12.   (3  +  4)x5-(4  +  10)-r-2-7. 

13.  (325  +  20)  -  (415  -  232)  -  47. 

14.  (532  -  40)  -  (315  -  116  +  7)  +  35. 

15.  (54-16)xlT+4-15x20. 

16.  [84-7x6  +  (3x5)-3]-9. 

17.  4  +  llx3-(5  +  28 


84 


DIVISION. 


RELATION    OP    DIVIDEND,    DIVISOR,    AND 
QUOTIENT. 

94.  The  value  of  the  quotient  depends  upon  that  of  the 
dividend  and  divisor.  If  one  of  these  is  changed,  while  the 
other  remains  the  same,  the  quotient  will  be  changed.  If 
both  are  changed,  the  quotient  may  or  may  not  be  changed. 

The  changes  may  be  illustrated  as  follows : 

FUNDAMENTAL    EQUATION. 
64  -5-  8  =  8. 


CHANGED    EQUATIONS. 

{'  1    19S       Q  _  1 «  ^i        1-  Multiplying  the  dividend 
1.  J^o  -^  c  by  2  multiplies  the   quotient 

I  by  2. 
2     32  -a-  8  =    4          2>  Dividing  the  dividend  by 
'  2  divides  the  quotient  by  2. 

1.  Multiplying  the    divisor 
by  2  divides  the  quotient  by 

2. 


2.  Divisor 
changed. 


3.  Both 
changed. 


1.  64  -f- 16  =    4 

2.  64  --   4  =  16 

1.  128  -r-  16  =  8 

2.  32--   4  =  8 


2.  Dividing  the  divisor  by 
2  multiplies  the  quotient  by  2. 

Multiplying  or  dividing  both 
dividend  and  divisor  by  2  does 
not  change  the  quotient. 


From  these  we  may  deduce  the  following  principles : 

95.  PRINCIPLES.  —  1.  Multiplying  the  dividend  or  dividing 
the  divisor  by  any  number,  multiplies  the  quotient  by  that 
number.' 

2.  Dividing  the  dividend  or  multiplying  the  divisor  by  any 
number,  divides  the  quotient  by  that  number. 

3.  Multiplying  or  dividing  both  dividend  and  divisor  by  the 
same  number,  does  not  change  the  quotient. 


ANALYSIS   AND  REVIEW.  85 

ANALYSIS   AND   REVIEW. 

96.  Analysis  is  the  process  of  solving  problems  by  tracing 
the  relation  of  the  parts. 

In  analyzing  it  is  usual  to  reason  from  the  given  number 
to  owe,  and  then  from  one  to  the  required  number. 


ORAL    EXERCISES. 

1.  If  8  yards  of  cloth  cost  $  16,  what  will  12  yards  cost  ? 

ANALYSIS.  —  Since  8  yards  cost  $  16,  1  yard  will  cost  one  eighth  of 
$16,  or  $2;  and  since  1  yard  costs  $2,  12  yards  will  cost  12  times 
$2,  or  $24. 

2.  If  5  coats  cost  $45,  what  will  7  coats  cost  ? 

3.  If  8  oranges  cost  32  cents^  what  will  9  oranges  cost  ? 

4.  A  man  bought  6  sheep  for  $36.     What  will  7  sheep 
cost  at  the  same  rate  ? 

5.  How  much  will  12  books  cost  if  8  books  cost  $  16  ? 

6.  If  9  tons  of  coal  cost  $  36,  what  will  be  the  cost  of 
7  tons  ? 

7.  If  8  barrels  of  flour  are  worth  $48,  what  are  12 
barrels  worth  ? 

8.  A  man  paid  $  160  for  4  cows.     How  much  would  he 
have  paid  had  he  bought  7  cows  at  the  same  rate  ? 

9.  What  must  be  paid  for  14  sheep,  if  5  sheep  cost  $  35  ? 

10.  In  8  hogsheads  there  are  504  gallons.     How  many 
gallons  do  9  hogsheads  contain  ? 

11.  If  12  men  can  do  a  piece  of  work  in  5  days,  how  long 
will  it  take  20  men  to  do  it  ? 


86  DIVISION. 

12.  If  it  requires  288  pickets  to  build  a  fence  9  rods  long, 
how  many  pickets  will  it  require  to  build  a  fence  15  rods 
long? 

13.  If  18  pounds  of  rice  are  worth  $1.44,  how  much  are 
20  pounds  worth  ? 

14.  In  20  quires  of  paper  there  are  480  sheets.     How 
many  sheets  are  there  in  8  quires  ? 

15.  If  a  man  receives  32  pounds  of  sugar  in  exchange  for 
20  pounds  of  cheese  at  8  cents  a  pound,  what  is  the  price  of 
the  sugar  per  pound  ? 

WRITTEN    EXERCISES. 

97.    1.    A  farmer  paid  $12,880  for  a  farm  of  112  acres. 
How  much  did  he  pay  for  8  acres  at  that  rate  ? 

SOLUTION.  112  acres  cost  $12,880. 

1  acre  costs  $  115. 
8  acres  cost  $920. 

2.  A  man  divided  $  41,185  equally  among  his  children, 
giving  to  each  $  8237.     How  many  children  did  he  have  ? 

3.  A  merchant  bought  175  yards  of  cloth  at  $6.50  per 
yard,  and  afterwards  sold  83  yards  at  $  7  per  yard,  and  the 
remainder  at  $  7.25  per  yard.     How  much  did  he  gain  ? 

4.  I  bought  at  a  store  12  pounds  of  sugar  at  5  cents  a 
pound,  3  quarts  of  syrup  at  15  cents  a  quart,  3  dozen  eggs 
at  24  cents  a  dozen,  4  pounds  of  butter  at  23  cents  a  pound, 
and  a  sack  of  flour  for  75  cents.     I  gave  the  salesman  a 
5-dollar  bill  to  pay  for  the  groceries.     How  much  change 
should  I  have  received  ? 

5.  A  dealer  bought  17  harrows  at  $  14  a  piece,  and  gave 
in  exchange  13  cords  of  wood  at  $  8  per  cord,  and  paid  the 
balance  in  cash.     How  much  cash  did  he  pay  ? 


WRITTEN   EXERCISES.  87 

6.  A  farmer  bought  7  oxen  at  $  65  each,  9  cows  at  $  42 
each,  and  120  sheep  at  $5.50  each.     What  did  he  pay  for 
the  whole  ? 

7.  How  many  yards  of  linen,  at  28  cents  a  yard,  must  be 
given  for  35  bushels  of  potatoes  at  56  cents  a  bushel  ? 

8.  Two  men  had  an  equal  interest  in  a  herd  of  cattle ; 
one  took  65  at  $  40  apiece,  and  the  other  took  the  rest  at 

apiece.     How  many  cattle  were  there  in  the  herd  ? 


9.  Mr.  A  has  a  sum  of  money  equal  to  8214  cents,  con- 
sisting of  an  equal  number  of  dollars,  dimes,  and  cents. 
How  many  has  he  of  each  ? 

10.  What  number  multiplied  by  123,  will  give  .a  product 
of  40,221  ? 

11.  Mr.  Faust  has  $  12,000  to  invest  in  land.     How  many 
acres  can  he  buy  at  $  125  an  acre  ? 

12.  Paid  $17,125  for  137  shares  of  railway  stock.     How 
much  did  each  share  cost  ? 

13  A  drover  had  580  sheep ;  he  sold  230  to  one  man,  213 
to  another,  and  then  bought  enough  to  make  his  number 
600.  How  many  did  he  buy  ? 

14.  Three  men  enter  into  partnership.     A  puts  in  $  2160 ; 
B,  $  1720  ;  and  C  twice  as  much  as  A  and  B  together.     How 
much  did  C  put  in,  and  how  much  did  all  together  invest  ? 

15.  Rome  was  founded  753  years   before  the  birth  of 
Christ.     How  long  is  it  since  that  time  ? 

16.  A  dying  man  willed  his  estate  as  follows :    To  his 
wife,  $4500;  to  each  of  his  three  sons,  $2800;  to  each  of 
his  five  daughters,  $  2500 ;  and  $  3000  to  a  church.     What 
was  the  value  of  the  estate  ? 


88  DIVISION. 

17.  I  bought  4  pairs  of  shoes  at  $  3.75  a  pair,  2  pairs  of 
boots  at  $  6.50  a  pair,  and  3  hats  at  $  2.75  each,  and  gave 
the  merchant  a  fifty-dollar  bilL     How  much  change  did  I 
get? 

18.  A  man  earns  $60  a  week,  and  spends  $  28  a  week. 
How  much  does  he  save  in  12  weeks  ? 

19.  Bought  13  cows  at  $  26  each,  9  horses  at  $  125  each, 
and  100  sheep  at  $  3.75  each,  and  sold  all  for  $  1750.     How 
much  did  I  lose  ? 

20.  Two  ships,  2500  miles  apart,  are  sailing  towards  each 
other,  one  at  the  rate  of  87  miles  a  day,  and  the  other  at 
the  rate  of  85  miles  a  day.     How  far  apart  will  they  be  at  the 
end  of  13  days  ? 

21.  Two  steamships  start  from  New  York  to  Liverpool, 
one  at  the  rate  of  156  miles  a  day,  the  other  at  the  rate  of 
217  miles  a  day.     How  far  apart  will  they  be  at  the  end  of 
9  days  ? 

22.  Mr.  L.  had  $100.     He  bought  a  suit  of  clothes  for 
$  22,  a  hat  for  $  3,  a  pair  of  shoes  for  $  4,  and  4  shirts  at 
$  1.75  each.     How  many  books  at  $  1.60  apiece  can  he  get 
for  the  remainder  of  his  money  ? 

23.  How  many  men,  at  a  salary  of  $  600  a  year,  will  earn 
$585,000  in  a  year? 

24.  It  requires  7,020,000  bricks  to  build  a  large  foundry. 
How  many  teams  will  it  require  to  draw  the  bricks  in  60 
days,  if  each  team  draws  6  loads  per  day  and  1500  bricks 
at  a  load  ? 

25.  From  a  cistern  containing  12,572  gallons  of  water, 
9236  gallons  were  drawn  out,  .and  afterwards,  during  a  rain- 
storm, 7250  gallons  ran  in.     How  much  water  was  there  then 
in  the  cistern  ? 


WRITTEN  EXERCISES.  89 

26.  I  bought  a  farm  of  163  acres  at  $  95  an  acre.     I  paid 
down  $  2500,  and  gave  bank-stock  valued  at  $  9000.     How 
much  remained  unpaid  ? 

27.  A   certain  cistern  holds   11,200   gallons   of    water. 
When  empty,  how  many  barrels  of  water,  each  containing 
31  gallons,  will  it  take  to  fill  it  ? 

28.  A  dairy-man  has  fodder  enough  to  keep  28  cows  4 
months.     If  he  sells  12  cows,  how  many  months  will  the 
fodder  last  the  rest  ? 

29.  If  the  difference  between  two  numbers  is  1320,  and 
the  smaller  number  is  1750,  what  is  the  larger  number  ? 

30.  The  sum  of  two  numbers  is  3680,  and  one  of  the 
numbers  is  1976.     What  is  the  other  number  ? 

31.  The  larger  of  two  numbers  is  2560,  and  their  differ- 
ence is  1177.     What  is  the  smaller  number  ? 

32.  What  is  the  product  of  14,625  and  12,349  ? 

33.  If  the  product  of  two  numbers  is  7,715,962,  and  one 
of  the  numbers  is  2566,  what  is  the  other  number  ? 

34.  I  paid  $  36  an  acre  for  50  acres  of  wood-land.    Isold 
the  wood  for  $  1576,  and  the  land  for  $  17  an  acre.     Did  I 
gain  or  lose,  and  how  much  ? 

35.  I  bought  a  stock  of  goods  for  $  12,650,  paying  $  1650 
cash,  and  the  balance  in  monthly  payments  of  $  1100  each. 
How  many  monthly  payments  did  I  make  ? 

36.  Twenty-four  ladies  and  18  gentlemen  went  on  an 
excursion,  and  their  expenses,  which  were  $  1.50  each,  were 
paid  by  the  gentlemen.    How  much  did  each  gentleman  pay  ? 

37.  A  young  man  worked  a  year  at  $35  a  month.     He 
paid  $16  a  month  for  his  board,  and  his  other  expenses 
amounted  to  $  96.     How  much  money  did  he  save  ? 


90  DIVISION. 

38.  A  stationer  paid  $  95  for  gold  pens,  at  $  1.90  apiece. 
How  many  pens  did  he  buy  ? 

39.  Find  the   cost  of  118  pounds  of  ham,   at   $.11   a 
pound ;  and  227  pounds  of  bacon,  at  $  .09  a  pound. 

40.  A  farmer's  wheat  crop  yielded  him  $565,  his  corn 
$  362,  his  oats  $  175,  and  his  clover  seed  $  57.     He  paid  a 
farm  hand  $  18  a  month  for  9  months,  $  225  for  fertilizers, 
and  $195.85  for  repairs  and  other  expenses.     How  much 
did  he  gain  on  his  farm  ? 

41.  Mr.  D.  invested  $3000  in  business.     The  first  year 
he  gained  $560,  and  the  second  year  he  lost  $475.     The 
next  three  years  his  average  yearly  gain  was  $  843.     How 
much  was  he  worth  at  the  end  of  the  five  years  ? 

42.  A  mechanic  receives  $1200  a  year  for  his  labor,  and 
his  expenses  are  $576.     In  how  many  years  can  he  save 
enough  to  buy  52  acres  of  land  at  $  72  an  acre  ? 

43.  A  man  paid  $450  for  a  horse  and  a  buggy,  the  horse 
being  valued  at  $  90  more  than  the  buggy.     What  did  he 
pay  for  each  ? 

44.  I  sold  a  quantity  of  wood  that  cost  $920  for  $  1265, 
thus  gaming  $3  a  cord.     How  many  cords  were  there  ? 

45.  B  bought  30  cows  for  $1400;  but  5  of  them  died. 
How  much  must  he  receive  for  each  of  the  rest  to  incur  no 
loss? 

46.  A  farmer  sells  125  acres  of  land  for  $  85  an  acre,  and 
75  acres  of  other  land  at  $  115  an  acre.     He  invests  all  the 
money  in  another  farm  that  costs  him  $  175  an  acre.     How 
many  acres  are  there  in  the  farm  ? 

47.  Light  travels  at  the  rate  of  about  186,000  miles  a 
second.     How  many  seconds  does  it  take  to  reach  us  from 
the  sun,  a  distance  of  about  95,000,000  miles  ? 


FACTOKS. 


98.   1.    What  is  the  product  when  7  is  multiplied  by  5? 
What  are  the  numbers  7  and  5  of  their  product  ? 

2.  What  numbers  multiplied  one  by  the  other  will  pro- 
duce 63  ?    What  are  the  numbers  7  and  9  of  63  ? 

3.  What  are  the  factors  of  a  number  ? 

4.  Name  the  factors  of  the  following  numbers :  20,  36, 
45,  48,  60,  35,  72,  50,  21,  40,  24,  32,  64,  81,  80,  56,  44. 

5.  What  numbers  will  exactly  divide  48  ?  40?  80?  81? 

6.  Since  6  exactly  divides  48,  what  part  of  48  may  it  be 
called  ? 

7.  Give  the  exact  divisors  of  the  following  numbers  :  40, 
81,  56,  42,  64,  32,  24,  50,  72,  35,  36,  45,  48,  60. 

8.  What  numbers   between  0  and   20   have   no   exact 
divisors  except  themselves  and  1  ?     What  name  is  given  to 
such  numbers  ?     Prime  Numbers. 

9.  What  are  the  prime  numbers  between  20  and  40  ? 

10.  What  numbers  between  0  and  40  have  exact  divisors 
besides  themselves  and  1  ?     What  name  is  given  to  such 
numbers  ?     Composite  Numbers. 

11.  Select  from  the  following  numbers,  first,  the  prime 
numbers ;  secondly,  the  composite  numbers :  13, 15,  21, 18, 27, 

91- 


92  FACTORS. 

23,  17,  40,  41,  37,  25,  19,  42,  47,  43,  20,  14,  28,  36,  35,  33, 
48,  49,  50,  51,  53,  72,  86,  66,  54,  80,  44,  71,  55,  63,  65,  67, 
77,  84,  81,  83,  61,  73,  88,  87,  99,  93,  92,  97. 

99.  A  number  that  expresses  whole  units  is  called  an 
Integer  or  Integral  Number. 

Thus,  18,  25,  30,  etc.,  are  integers  or  integral  numbers. 

100.  The  integers  which  multiplied  by  one  another  will 
produce  a  number  are  called  the  Factors  of  the  number. 

Thus,  7  and  5  are  the  factors  of  35. 

101.  An  integer  that  will  divide  a  number  without  having 
a  remainder  is  called  an  Exact  Divisor  of  the  number. 

Thus,  2,  3,  4,  and  6  are  exact  divisors  of  12. 

The  factors  of  a  number  are  exact  divisors  of  it. 

102.  A  number  that  has  no  exact  divisor  except  itself 
and  1  is  called  a  Prime  Number. 

Thus,  1,  3,  5,  7,  11,  13,  etc.,  are  prime  numbers. 

103.  A  number  that  has  exact  divisors  besides  itself  and 
1  is  called  a  Composite  Number.     Hence,  a  composite  num- 
ber is  always  the  product  of  two  or  more  factors. 

Thus,  12,  18,  21,  40,  etc.,  are  composite  numbers. 

104.  A  number  that  is  exactly  divisible  by  2  is  called  an 
Even  Number. 

Thus,  10,  12,  16,  18,  etc.,  are  even  numbers. 

105.  A  number  that  is  not -exactly  divisible  by  2  is  called 
an  Odd  Number. 

Thus,  3,  5,  9,  11,  13,  etc.,  are  odd  numbers. 


TESTS  OF   DIVISIBILITY.  98 

/ 
TESTS  OP  DIVISIBILITY. 

106.   Illustrate  with  numbers  the  truth  of  each  of  the 
following  statements : 

1.  Two  is  an  exact  divisor  of  any  number  whose  right- 
hand  digit  is  0,  2,  4,  6,  or  8. 

2.  Three  is  an  exact  divisor  of  any  number,  the  sum  of 
whose  digits  is  divisible  by  3. 

3.  Four  is  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  two  right-hand  digits  is  divisible  by  4. 

4.  Five  is  an  exact  divisor  of  any  number  whose  right- 
hand  digit  is  0  or  5. 

5.  Six  is  an  exact  divisor  of  any  even  number,  the  sum 
of  whose  digits  is  divisible  by  3. 

6.  Eight  is  an  exact  divisor  of  a  number,  if  the  number 
expressed  by  its  three  right-hand  digits  is  divisible  by  8. 

7.  Nine  is  an  exact  divisor  of  any  number,  the  sum  of 
whose  digits  is  divisible  by  9. 

8.  Twenty-five  is  an  exact  divisor  of  a  number,  if  the  num- 
ber expressed  by  its  two  right-hand  digits  is  divisible  by  25. 

9.  One  hundred  twenty-five  is  an  exact  divisor  of  a  num- 
ber, if  the  number  expressed  by  its  three  right-hand  digits 
is  divisible  by  125. 

10.  If  an  even  number  is  divisible  by  an  odd  number,  it 
is  divisible  by  twice  that  number. 

11.  An  exact  divisor  of  a  number  is  an  exact  divisor  of 
any  number  of  times  that  number. 

12.  An  exact  divisor  of  each  of  two  numbers  is  an  exact 
divisor  of  their  sum  and  of  their  difference. 


94 


FACTORS. 


EXERCISES. 

107.   Find  by  inspection  some  of  the  exact  divisors  of  the 
following  numbers : 


1. 

43844. 

9. 

72369. 

17. 

83466. 

25. 

27324. 

2. 

39128. 

10. 

85728. 

18. 

48324. 

26. 

34254. 

3. 

46836. 

11. 

51650. 

19. 

51375. 

27. 

38655. 

4. 

37125. 

12. 

43284. 

20. 

48224. 

28. 

51750. 

5. 

48636. 

13. 

31296. 

21. 

31959. 

29. 

62488. 

6. 

41848. 

14. 

53712. 

22. 

40675. 

30. 

83830. 

7. 

36444. 

15. 

48375. 

23. 

58625. 

31. 

71127. 

8. 

52146. 

16. 

31475. 

24. 

38169. 

32. 

92625. 

FACTORING. 

108.  The  process  of  separating  a  number  into  its  factors 
is  called  Factoring. 

109.  Factors  that  are  prime  numbers  are  called  Prime 
Factors. 

Thus,  5  and  7  are  the  prime  factors  of  35. 

110.  When  numbers  have  no  common  factor  they  are 
said  to  be  Prime  to  Each  Other. 

Thus,  7  and  16  are  prime  to  each  other,  though  16  is  not  a  prime 
number. 

111.  The  number  of  times  a  number  is  used  as  a  factor 
is  indicated  by  a  small  figure  called  an  Exponent. 

It  is  written  above  and  at  the  right  of  the  number. 

Thus,  5  used  as  a  factor  4  times  is  indicated  by  54. 


CANCELLATION.  95 

1.    What  are  the  prime  factors  of  1008  ? 


2 

1008 

EXPLANATION        SincG  GVGrv  Drinif1  "fjiptov  of  <\>  TinTn'V 

er 

2 

504 

is  an  exact  divisor  of  the  number,  the  prime  factors 

of 

2 

252 

1008  may  be  found  by  finding  all  the  prime  numbers 

2 

126 

that  are  exact  divisors  of  1008.     Since  the  number 

is 

even,  2  is  taken  for  the  first  prime  divisor.     Since  the 

/*  O 

00 

quotient  is  even,  2  is  taken  for  divisor  again,  and  the 

3 

9 

division  is  continued  until  the  last  quotient  is  1. 

3 

3 

Hence  the  prime  factors  are  2,  2,  2,  2,  7,  3,  3  or  24, 

7, 

1 

32. 

RULE.  — 

Divide  the  given  number  by  any  prime  number 

that  will  exactly  divide  it.     Divide  this  quotient  by  another 

prime  number,  and  so  continue  until  the  quotient  is  1. 

The  several  divisors  will  be  the  prime  factors. 

What  are 

the  prime  factors  of  the  following  : 

2 

.     45. 

10.     360.           18.     4862.          26.     21504. 

3 

.     84. 

11.     786.           19.     3290.          27.     10010. 

4 

.   125. 

12.  1872.           20.     4620.          28.     32320. 

5 

.   210. 

13.  2310.           21,     3136.          29.     25600. 

6 

.   315. 

14.  3465.           22.     3812.          30.     64384. 

7 

.   432. 

15.  2205.           23.     7007.          31.     31570. 

8 

.   330. 

16.  6300.           24.     4350.          32.     48500. 

9 

.    484. 

17.  7644.           25.  11368.          33.  124416. 

CANCELLATION. 

112.  1.  How  many  times  is  8  times  5  contained  in  16 
times  5  ?  4  times  12  in  16  times  12  ?  5  times  7  in  15 
times  7  ? 

2.  How  many  times  is  9  x  8  contained  in  27  x  8  ?  8x6 
in  24  x  6  ?  15  x  9  in  30  x  9  ?  4  x  18  in  12  x  18  ? 


96  FACTORS. 

3.  In  determining  the  quotient,  what  numbers  may  be 
omitted  from  both  dividend  and  divisor  ? 

113.  The  process  of  shortening  computations  by  reject- 
ing equal  factors  from  both  dividend  and  divisor  is  called 
Cancellation. 

114.  PRINCIPLE.  —  Rejecting  equal  factors  from  both  divi- 
dend and  divisor  does  not  alter  the  quotient. 

WRITTEN    EXERCISES. 

115.  1.    Divide  6  x  12  x  15  x  36  by  3  x  4  x  5  X  48. 

0333  EXPLANATION. — The  dividend 

&  v  to1  v  IK  v  £<T?      97  is  written  above  the  divisor  with 

p  A   LAt   A   La  A  <py>        *H          -\r>-i  MI 

g ^ ~ ^rr  =  -9-  =  13J.      a  line  between  them. 

p  X    £  X    pXw  Since  3^  4^  and  5  are  factors  of 

6,  12,  and  15,  respectively,  in  the 
*  dividend,  they  may  be  rejected 

from  both  dividend  and  divisor,  leaving  the  factors  2,  3,  and  3  in 
the  dividend.  Since  12  is  a  factor  of  36  in  the  dividend  and  of  48  in 
the  divisor,  it  may  be  rejected,  leaving  3  in  the  dividend  and  4  in  the 
divisor.  Since  2,  one  of  the  factors  left  in  the  dividend,  is  a  factor  of 
the  4  left  in  the  divisor,  it  may  be  rejected,  leaving  2  in  the  divisor. 

The  product  of  the  factors  not  canceled  in  the  dividend  is  27,  and 
of  those  in  the  divisor,  2.  Hence,  the  quotient  is  V-,  or  27  -=-  2,  or  13£. 

RULE.  —  Reject  from  the  dividend  and  divisor  all  factors 
common  to  both,  and  then  divide  the  product  of  the  remaining 
factors  of  the  dividend  by  the  product  of  the  remaining  factors 
of  the  divisor. 

When  all  the  factors  of  both  dividend  and  divisor  are  canceled,  the 
quotient  is  1,  for  the  dividend  then  contains  the  divisor  exactly  once. 

Divide,  using  cancellation: 

2.  6  x  9  x  12  x  15  x  20  by  3  x  3  x  4  x  5  x  30. 

3.  5  x  8  x  24  x  30  by  4  x  3  x  5  x  10. 

4.  7x6x5x4x3  by  8x3x5. 

5.  36  x  24  by  8  x  4  x  6. 


CANCELLATION.  97 

s 

6.  45  x  30  x  9  by  5  x  9  x  50. 

7.  12  x  6  x  9  x  20  by  4  x  80  x  3. 

8.  2x3x5x7x9xllby4x6xl8x7. 

9.  14  x  9  x  7  x  15  x  21  by  42  x  3  x  7. 

10.  17  x  14  x  13  x  12  by  7  x  3  x  26  x  34. 

11.  27  x  49  x  38  x  25  by  35  x  18  x  15. 

12.  28  x  54  x  72  by  14  x  9  x  36. 

13.  114  x  85  x  75  by  15  x  5  x  57  x  17. 

14.  140  x  65  x  27  by  13  x  20  x  9. 

15.  95  x  66  x  81  by  9  x  3  x  11  x  19. 

16.  78  x  14  x  63  x  5  by  7  x  13  x  7  x  21. 

17.  69  x  37  x  28  x  45  by  15  x  23  x  7  x  3. 

18.  144  x  82  x  49  by  7  x  2  x  12  x  41. 

19.  57  x  148  x  64  by  36  x  19  x  4. 

20.  84  x  96  x  108  by  27  x  14  x  12. 

21.  117  x  57  x  49  by  114  x  7  x  13. 

22.  121  x  8  x  90  by  4  x  10  x  11  X  2. 

23.  216  x  33  x  72  by  18  x  11  X  36. 

24.  45  x  56  x  68  x  92  by  7  x  9  x  46. 

25.  75  x  125  x  33  x  28  by  14  x  16  x  150. 

26.  Divide  the  product  of  47  times  27  times  45,  by  9 
times  81. 

27.  Find  the  quotient  of  77  times  360  times  475,  divided 
by  11  times  6  times  35. 

28.  How  many  bushels  of  corn,  worth  55  cents  a  bushel, 
must  be  given  in.  exchange  for  3  pieces  of  cloth,  each  con- 
taining 33  yards,  at  25  cents  a  yard  ? 

29.  How  many  boxes  of  coffee,  each  containing  40  pounds, 
at  28  cents  a  pound,  must  be  given  for  30  firkins  of  butter*, 
each  containing  56  pounds,  at  18  cents  a  pound  ? 

STAND    AR.  —  7 


98  FACTORS. 

30.  How  many  barrels  of  flour,  at  $  6  a  barrel,  must  be 
given  for  3  pieces  of  linen,  each  containing  36  yards,  at 
25  cents  a  yard  ? 

31.  How  many  bushels  of  wheat,  at  90  cents  a  bushel, 
will  pay  for  3  barrels  of  sugar,  each  containing  200  pounds, 
at  6  cents  per  pound  ? 

32.  A  laborer  worked  8  days  for  24  bushels  of  potatoes, 
worth  40  cents  a  bushel.     What  were  his  daily  earnings  ? 

33.  How  many  boxes  of  tea,  each  containing  24  pounds, 
worth  45  cents  a  pound,  must  be  given  for  4  loads  of  wheat, 
each  containing  54  bushels,  worth  $  .95  a  bushel  ? 

34.  If  34  bushels  of  wheat  make  8  barrels  of  flour,  how 
many  bushels  will  be  necessary  to  make  72  barrels  ? 

35.  A  grocer  sold  18  boxes  of  soap,  each  containing  55 
pounds,  at  10  cents  a  pound,  and  received  as  pay  66  barrels 
of  apples,  each  containing  3  bushels.     What  was  the  price 
per  bushel  of  the  apples  ? 

36.  A  fanner  exchanged  96  bushels  of  corn,  worth  $  .55 
a  bushel,  for  an  equal  number  of  bushels  of  rye,  worth  $.75 
a  bushel,  and  oats  worth  $  .35  a  bushel.    How  many  bushels 
of  each  did  he  receive  ? 

37.  If  48  men  can  dig  a  trench  in  25  days,  working  9 
hours  a  day,  how  many  days  will  be  required  by  20  men  to 
do  the  same  work  if  they  work  10  hours  per  day  ? 

38.  If  12  barrels  of  pork,  each  containing  200  pounds, 
are  worth  $  192,  what  will  80  pounds  cost  at  the  same  rate 
per  pound  ? 

39.  How  many  days'  work,  at  $  1.25  a  day,  will  pay  for 
75  bushels  of  corn,  at  $  .60  cents  a  bushel  ? 


PEACTIONS. 


116.  1.    When  anything  is  divided  into  two  equal  parts, 
what  is  each  part  called  ?     Into  three  equal  parts  ?     Into 
seven  equal  parts  ?     Into  eight  equal  parts  ?     Into  nine 
equal  parts  ?     Into  fifteen  equal  parts  ? 

2.  How  many  halves  are  there  in  anything?     How  many 
thirds  ?     How  many  fifths  ?     How   many  tenths  ?     How 
many  fifteenths  ?     How  many  twentieths  ? 

3.  What  part  of  an  apple  will  each  boy  receive  when  it 
is  divided  equally  among  7  boys  ?     Among  8  boys  ? 

4.  How  much  is  one  fifth  of  10  cents  ?    Of  15  cents  ?    Of 
30  cents  ? 

5.  How  much  is  one  sixth  of  12  cents?     Of  18  cents? 
Of  24  cents  ? 

6.  How  much  is  one  seventh  of  14   oranges  ?      Of  21 
oranges  ?     Of  28  oranges  ? ' 

7.  In  6  hours  James  earned  36  cents.      How  much  did 
he  earn  per  hour  ? 

8.  A  school  has  4  classes  of  the  same  size,  and  the  whole 
number  of  pupils  in  the  classes  is  40.     How  many  are  there 
in  each  class  ? 

117.  One  or  more  of  the  equal  parts  of  anything  is  called  ' 
a  Fraction. 

Two  numbers,  written  one  above  the  other  with  a  line 
between  them,  are  used  to  express  a  fraction. 

99 


100  FRACTIONS. 

118.  The  number  which  shows  into  how  many  parts  a 
thing  has  been  divided  is  called  the  Denominator. 

It  is  written  below  the  line. 

Thus,  in  the  fraction  |,  9  is  the  denominator.  It  shows  that  some- 
thing has  been  divided  into  9  equal  parts. 

119.  The  number  which  shows  how  many  parts  form  the 
fraction  is  called  the  Numerator. 

It  is  written  above  the  line. 

Thus,  in  the  fraction  $,  7  is  the  numerator.  It  shows  that  the  frac- 
tion contains  7  of  the  9  equal  parts. 

120.  The  numerator  and  denominator  together  are  called 
the  Terms  of  the  Fraction. 

121.  A  fraction  whose  numerator  is  less  than  its  denomi- 
nator is  called  a  Proper  Fraction. 

Thus,  f ,  f,  and  |f  are  proper  fractions. 

The  value  of  a  proper  fraction  is,  therefore,  less  than  1. 

122.  A  fraction  whose  numerator  equals  or  exceeds  its 
denominator  is  called  an  Improper  Fraction. 

Thus,  |,  -1/-,  and  f  1  are  improper  fractions. 

The  value  of  an  improper  fraction  is,  therefore,  1  or  more  than  I.     • 

123.  A  number  expressed  by  an  integer  and  a  fraction 
is  called  a  Mixed  Number. 

Thus,  3f,  8|,  and  14 1|  are  mixed  numbers. 

124.  The  unit  which  is  divided  into  equal  parts  is  called 
the  Unit  of  the  Fraction. 

A  fraction  whose  unit  has  been  divided  into  any  number  of  equal 
parts  is  called  a  Common  Fraction. 

A  fraction  whose  unit  has  been  divided  into  tenths,  hundredths, 
thousandths,  etc.,  is  called  a  Decimal  Fraction. 


DEFINITIONS.  101 

125.  One  of  the  equal  parts  into  which  a  unit  has  been 
divided  is  called  a  Fractional  Unit. 

126.  A  fraction  also  expresses  unexecuted  division,  the  nu- 
merator being  the  dividend  and  the  denominator  the  divisor. 

Thus,  ty  is  equal  to  10  -=-  5  ;  \9-  is  equal  to  19  -r-  4. 

127.  Fractions  are  read  by  naming  first  the  number  of 
fractional  units,  and  then  the  kind  of  them. 

Thus,  £  is  read  three  eighths  ;  ¥9r  nine  thirty-firsts. 

128.  Eead  the  following : 

*       !     A     tf     H     to     If     H     tt     » 
A     A     A     A-     it     «     «     ft     «     tt 

Ttfo"       ¥T9"       "FIT       "STG"       TT9        TT9"       T3~9        5T8"       TFS"        6"T? 

Express  in  figures : 

1.  Seven  ninths.     Eight  elevenths.     Four  fifteenths. 

2.  Six  nineteenths.   Nine  fourteenths.    Five  seventeenths. 

3.  Three  twentieths.    Six  forty-fifths.    Nine  eighteenths. 

4.  Seventeen  twentieths.     Thirteen  forty-sevenths. 

5.  Twelve  thirtieths.     Fifty-five  eighty-fifths. 

6.  Seventeen  fifty-fifths.     Thirty-four  ninety-eighths. 

129.  1.    Interpret  the  expression  -£. 

EXPLANATION.  —  |  represents  7  of  9  equal  parts  of  a  thing.    It  also 
represents  one  ninth  of  7,  and  7  divided  by  9.    It  is  read  seven  ninths. 

In  like  manner  interpret  the  following : 

2-  i    A    H    tt    «    «    tt    n 

3-  AAtttttttttttt 


102  FRACTIONS. 

REDUCTION. 
130.    To  reduce  fractions  to  higher  terms. 

1.  In  \  yard,  how  many  fourths  are  there  ?    How  many 
eighths  ? 

2.  In  \  of  a  foot,  how  many  sixths  are  there  ?     How 
many  ninths  ? 

3.  Since  \  is  equal  to  f,  how  may  the  terms  of  the 
fraction  f  be  obtained  from  \  ?     f  from  |  ?     f  from  £  ? 
f  fromi? 

4.  Since  the  terms  of  the  fraction  f  may  be  obtained 
from  ^  by  multiplying  them  by  4,  how  may  the  terms  of  the 
fraction  %  be  obtained  from  |  ? 

5.  What  changes,  then,  may  be  made  in  the  terms  of  a 
fraction  without  changing  its  value? 


Change  the  following  : 

6. 

f 

to 

12ths. 

13. 

t 

to 

18ths. 

20. 

* 

to 

18ths. 

7. 

t 

to 

20ths. 

14. 

5 

"8 

to 

24ths. 

21. 

f 

to 

25ths. 

8. 

f 

to 

14ths. 

15. 

* 

to 

27ths. 

22. 

t 

to 

16ths. 

9. 

t 

to 

16ths. 

16. 

1 

to 

24ths. 

23. 

f 

to 

14ths. 

10. 

t 

to 

21sts. 

17. 

f 

to 

18ths. 

24. 

1 

to 

25ths. 

11. 

f 

to 

15ths. 

18. 

t 

to 

20ths. 

25. 

1 

to 

18ths. 

12. 

1 

to 

18ths. 

19. 

f 

to 

21sts. 

26. 

t 

to 

30ths. 

131.  The  process  of  changing  the   forms  of   fractions 
without  changing  their  values  is  called  Reduction  of  Frac- 
tions. 

132.  A  fraction  is  expressed  in  higher  terms,  when  its 
numerator  and  denominator  are  expressed  by  larger  numbers. 


REDUCTION. 


103 


133.  PRINCIPLE.  —  Multiplying  or  dividing  both  terms  of 
G,  fraction  by  the  same  number  does  not  change  the  value  of 
the  fraction. 

WRITTEN   EXERCISES. 

134.  1.    Change  T7F  to  48ths. 


48  -f-  16  =  3 
7  X  3  =  21 
16  X  3  =  48 


EXPLANATION.  —  Since  there  are  48  forty-eighths  in 
i}  in  ^  there  are  ^  Of  |f  ,  or  ^  ;  and  in  ^  there  are 
7  times  ^  or  f  i.  Or, 

Since  the  denominator  of  the  fraction  is  to  be  48, 
both  terms  of  the  fraction  must  be  multiplied  by  3. 

RULE.  —  Multiply  the  terms  of  the  fraction  by  such  a  num- 
ber as  will  change  the  given  denominator  to  the  required 
denominator. 


Change  the  following 
2.    |f  to70ths.        9. 
3.    If-  to  54ths.      10. 
4.    |fto64ths.      11. 
5.   |f  to  58ths.     12. 
6.    |f  to  78ths.      13. 
7.    |f  to  93ds.       14. 
8.    |f  to  84ths.      15. 

» 

n 

« 

to 
to 
to 
to 
to 
to 
to 

108ths. 
135ths. 
216ths. 
165ths. 
lllths. 
144ths. 
168ths. 

16. 
17. 
18. 
19. 
20. 
21. 
22. 

ft 
« 

ft 

to 
to 
to 
to 
to 
to 
to 

212ths. 
ISOths. 
219ths. 
236ths. 
152nds. 
196ths. 
355ths. 

135.   To  reduce  fractions  to  lower  terjns. 

1.  How  many  halves  are  there  in|?     f?  -f$ 

2.  How  many  thirds  are  there  inf?     f?  -^ 

Reduce  the  following  to  lower  terms : 

3-  A   A   y9*  A   tt  *  tt  A- 

*-AAAAAAA  A- 

s-  A   A   A   A   T%   A   A  A- 

^  A  A  A  A  A  A  A  A- 


104  FRACTIONS. 

136.  A  common  divisor  of   two  or  more  numbers  is  a 
number  that  will  exactly  divide  each  of  them. 

137.  A  fraction  is  expressed  in  lower  terms  when  its 
numerator  and  denominator  are  expressed  in  smaller  num- 
bers. 

A  fraction  is  expressed  in  its  lowest  terms  when  its 
numerator  and  denominator  have  no  common  divisor. 

WRITTEN    EXERCISES. 

138.  1.    Reduce  |f  to  its  lowest  terms. 

5)45 9  EXPLANATION.  —  Since  the  fraction  is  to  be  reduced 

g\  gQ  ^[2  to  its  lowest  terms,  the  terms  are  first  divided  by  5 

(Prin.,  Art.  133),  and  the  terms  of  the  resulting  frac- 
3)  9  _  3  tion  by  3.  Inasmuch  as  no  number  will  exactly  divide 
3\  ^2  =  4  tne  terms  of  the  fraction  f ,  the  fraction  is  reduced  to 

its  lowest  terms. 

RULE.  —  Divide  the  terms  of  the  fraction  by  any  common 
divisor,  and  continue  thus  to  divide  until  the  terms  have  no 
common  divisor.  See  also  Art.  511. 

Reduce  to  their  lowest  terms : 
2.  if,  |f        12.  |ffc  iff.        22.  Iff,  ffo.        32.  iff,  ff£. 

q      3  6.     63  10       115     1 05  oo       272     19  5  qq       363     504 

'    T2>  "g"¥*          J>0>    2T3>  T2T'          /5°*    T2~5>  T87'          °°'    "60T>  "STir 

4.     48      29  ^4^     121      2.4.5^          £4.     216     182^          34^    ^00      64  8^ 

5-  IS,  «•        15-  iff,  flfc.        25.   ^  |ff.        35.   ||f,  f|f. 

6-  It.  «•        16.   iff,  i%.        26.   i|f,  |1J.        36.   |J(,  |ff 

7-  f},  |fr        17-   iff.  -«•        27.  iff,  tff        37.   |ff,  fff. 

»•  4*.  it-    is.  HI,  «f    28.  iff,  nf    ss.  m,  m- 

9-  ft,  «•  19-  JH,  «|-  29.  Hf,  |ff  39.  ffj,  |f|. 
1°-  M.  A-  20.  i|J,  iff.  30.  |ff,  |ff.  40.  f|f,  fff. 
11-  M.  *  2!.  Iff,  iff.  31.  iff,  ffj.  41.  Hi,  |f|. 


REDUCTION.  105 

139.  To  reduce  integers  and  mixed  numbers  to  improper 
fractions. 

1.  How  many  fourths  are  there  in  an  orange?     In  2 
oranges  ? 

2.  How  many  sixths  are  there  in  a  cake  ?     In  3  cakes  ? 

3.  How  many  fifths  are  there  in  1  ?  In  2  ?  In  3  ?  In  4  ? 

4.  How  many  eighths  are  there  in  1  ?    In  2  ?    In  5  ? 
In  8  ?     In  9  ?     In  10  ? 

5.  How  many  sevenths  are  there  in  1  ?    In  1±  ?    In  2  ? 
In  2f  ?     In  6  ? 

Reduce  the  following  to  improper  fractions  : 

a       91  Q2  A1  K3  £2  71 

D.      Air  Ofr  O-g-  I  -g-. 

9.    51  3f  4f  2$  9J  3-J-. 

10.  2|  6£  6^-  5|  4|  7f. 

11.  3f  5f  5J  2f  2|  6f 

WRITTEN    EXERCISES. 

140.  1.    Reduce  18f  to  an  improper  fraction. 

18  =  -9-=S-          EXPLANATION.  —  Since  in  1  there  are  5  fifths,  in  18 

90       3  _  93      there  are  18  times  5  fifths'  or  -9/  ;  and  in  18  +  f  there 
~F"  ~H  t  =  ~s~     are  -9/-  -f  f ,  or  -953-.     Hence,  18|  is  equal  to  -9/. 

RULE.  —  Multiply  the  integer  by  the  given  denominator,  to 
this  product  add  the  numerator,  and  write  the  result  over  the 
given  denominator. 


106  FRACTIONS. 

Reduce  the  following  to  improper  fractions : 


2. 

18f 

8. 

36«. 

14. 

421H- 

20. 

526£f 

3. 

34f 

9. 

48||. 

15. 

347H- 

21. 

635|f. 

4. 

36^. 

10. 

51^- 

16. 

423ff 

22. 

717i|. 

5. 

47|f 

11. 

68ft 

17. 

450ff. 

23. 

694ff 

6. 

63J|. 

12. 

72|f 

18. 

479£f 

24. 

800ff 

7. 

60|f 

13. 

89ff 

19. 

399ff 

25. 

815T\\ 

26.    Change  24  to  5ths;  13  to  8ths;  45  to  7ths;  37  to 
12ths ;  37  to  15ths ;  24  to  17ths ;  36  to  25ths ;  41  to  40ths. 

141.  To  reduce   improper  fractions  to  integers   or  mixed 
numbers. 

1.  To  how  many  dollars  are  8  quarter-dollars  equal  ?    16  ? 
20?     25?     30? 

2.  To  how  many  bushels  are  12  fourths  of  a  bushel  equal  ? 
20?     25?     50? 

3.  How  many  units  are  there  in  5  fifths  ?     In  10  fifths  ? 
In  15  fifths  ? 

Reduce  to  integers  or  mixed  numbers : 

4.  ±f-    2£-    -34-    4-5-    -2-9-    -5-4    -5A  • 

5.  U-        32.       43        47       _4_1_       _5_3_       49 

WRITTEN    EXERCISES. 

142.  1.    Reduce  -^-jp-  to  a  mixed  number. 

EXPLANATION.  —  Since  9  ninths  are  equal 

8g6 OKC   .  Q DOS       t°  1  unit,  356  ninths  are  equal  to  as  many 

7      units  as  9  ninths  are  contained  times  in  366 
ninths,  or  39f.     Therefore,  ^  =  39f. 


REDUCTION.  107 

EULE. — Divide  the  numerator  by  the  denominator. 
Eeduce  to  integers  or  mixed  numbers  : 

2.  9-8      JL2..  9.      2_2^     1^8  >  -±Q       84^  £3.      849|6. 

3.  ||;  |f,        10.    ^,  -23^L.          17.    £ff£.          24.    ±|f4*. 

4.  ^  ff-      11-  W/W-       18-  -Hf1-       25-  ^ffi4-- 

5.  ff,  f|.  12.  ^f,  *ff.-  19'  HF-  26« 

6.  f|,  |f.  13.  -^,  ^.  20.  lf£4.  27. 

7.  Ifj  |l,  14.  J^,  Jg^L.  21.  ^fp.  28. 

8.  f|,  |f.  15.  %8->  %*-•  22-  ^W1-  29' 

143.    To  reduce  dissimilar  fractions  to  similar  fractions. 

1.  How  many  eighths  are  there  in  £  ?     In  J  ? 

2.  How  many  sixths  are  there  in  £  ?     In  ^  ?     In  |  ? 

3.  How  many  twelfths  are  there  in  J?     In  £?     In 

4.  Into  what  parts,  then,  of  the  same  size,  may  f,  -^, 
and  -J-  be  divided  ? 

5.  Into  what  parts  of  the  same  size  may  %  and  \ 
divided?     \  and  -|-?     \  and  y1^?     £  and  -|-?     £  and  -^ 
£,  £,  and  £  ?     -J-,  ^,  -J-,  and 


Eeduce  the  following  to  fractions  having  the  same  de- 
nominators : 

6.  £  and  |.  11.  f  and  f.  16.  •£,  |,  and  T\. 

7.  Jaadf  12.  f  and  f  17.  i  |,  and  T%. 

8.  iandf  13.  fandf  18.  J,  f ,  and  f 

9.  Jandf.  14.  fandf.  19.  |,  ±,  and  f. 
10.   ^andf  15.  fandf.  20.  ±,  $,  and  ^. 


108  FRACTIONS. 

144.  Fractions   that   have  the   same   denominators   are 
called  Similar  Fractions. 

145.  Fractions  that  have  not  the  same  denominators  are 
called  Dissimilar  Fractions. 

146.  The  denominator  of  similar  fractions  is  called  a 
Common  Denominator. 

147.  When  similar  fractions  are  expressed  in  their  lowest 
terms,  they  have  their  Least  Common  Denominator. 

WRITTEN   EXERCISES. 

148.  1.    Reduce  f,  f,  and  -J-J,  to  similar  fractions. 

3  _  3x6  18  EXPLANATION.  —  Since  the  fractions  are  to  be 
J  =  4  x  £  ==  24  cnanSed  to  other  fractions  having  a  common 
denominator,  the  terms  of  each  fraction  must 
_  ==  £  X  o  _  lo  ke  muitiplied  by  some  number  which  will  cause 
8  8x3  24  them  to  have  the  same  denominator.  (Prin., 

11_11  X2_22     Art-  133-) 

T^  —  :rx      r>  —  7^7         By  examining  the  denominators  4,  8,  and  12, 
X  it  is  evident  that  the  denominators  of  all  the  frac- 

tions can  be  made  24,  and  the  fractions  will  then  be  similar.  To  make 
the  denominators  24,  the  terms  of  the  first  fraction  must  be  multiplied 
by  6  ;  the  terms  of  the  second,  by  3  ;  the  terms  of  the  third,  by  2.  And 
thus,  the  fractions  are  changed  to  the  similar  fractions  ||,  |f,  f  f. 

Keduce  to  similar  fractions : 

2.    |,  |,  A.  10.    f,  |,  TV  18.    J,  4,  1,   iV 

3-    f,  t  f  11.    f,  *,  iV  19-    i,  i  A»  A- 

4.  i,  |,  f.  12.  T\,  A,  A-     20-  f  A>  A»  i- 

5.    |,  |,  ^.  13.    |,  A,  A-          21.    T%,  |,  ^,  Jfc, 

6-  1%,  A*  A-  14-    *»    &>  f  22.    f,  TV  ^  f 

7-  I,  A?   254-  15-    *,  A»  A-          23.    f,  /T,  ^  _5¥. 

8.  |,  |,  A-  16-    A,  i  A-          24.    f,  .&,  TV  f 

9.  A,  A,  f  17.    A,  |,  H-          25.    f,  A,  A,  ^ 


REDUCTION.  109 

26.    Reduce  -|,  J,  T7^,  and  T9^-  to  similar  fractions  having 
their  least  common  denominator. 


3     4     12     16          EXPLANATION.  —  The  least  common  denomi- 


1      4       4     16     nator  cannot  always  be  easily  found  by  inspec- 
tion.   It  may  then  be  found  as  in  the  margin. 


1228 


Since  the  least  common  denominator  must  be 
1114     the  smallest  number  that  will  contain  each  of 
the  denominators,  it  must  contain  each  of  the 

3x2x2x4  =  48     prime  factors  of  the  denominators  and  no  other 
factors.    The  prime  factors  are  found  as  in  the 

margin.  3  is  a  prime  factor  of  3  and  12,  and  consequently  a  factor  of 
the  least  common  denominator.  Dividing  by  3,  and  writing  below 
the  quotients  and  numbers  of  which  3  is  not  a  factor,  we  have,  1,  4, 4, 16. 
Dividing  by  2,  and  again  by  2,  the  factors  of  the  denominator  are 
found  to  be  the  divisors  3,  2,  2,  and  the  factor  4  in  the  last  row.  Their 
product  is  48,  the  least  common  denominator.  The  fractions  thus 
become  ff ,  f f ,  f f,  |f • 

NOTE.  —  Fractions  should  first  be  reduced  to  their  lowest  terms.  In 
finding  the  factors  of  the  least  common  denominator  a  number  that  is 
a  factor  of  another  number  may  be  disregarded.  Thus,  since  3  and  4 
are  factors  of  12,  they  might  have  been  disregarded,  and  the  factors  of 
12  and  16  only  found.  See  also  pages  389-391. 

Change  to  similar  fractions  having  their  least  common 
denominator : 

27-  },  f,  |,  A-  37.  f,  H>  &>  If 

28.  i,  f,  |,  f  38.  f,  «,  |f,  if. 

29-  i,  I,  I,  A-  39.  |,  T5T,  ft,  ^. 

30.  |,  |,  A>  «•  40.  £-,  A»  if,  A- 

31  •  T3TJ    TTT?    TO"'    TO"'  41*  5?    3lT>    T5>   TS' 

32.  fc  fc  |,  /p  42.  if,  if,  it  f 

33.  f,  A  ii  «.  43.  £,  A,  if,  A- 

34.  1,  t,  |,  f.  44.  -H,  A,  $$,  ^ 

35-    A,  A,  A»  tf-  45>    4t>   6^  H>  TV 

36.    A,  /p  M,  «.  46.    7|,  9f,  8 


FRACTIONS. 


ADDITION. 

149.  1.  James  spent  £  of  a  dollar  for  an  arithmetic, 
-£  for  a  slate,  and  %  for  a  geography.  How  much  did  he 
spend  for  all  ? 

2.  I  bought  %  of  a  yard  of  silk,  but  afterward  I  was 
compelled  to  buy  f  of  a  yard  more.     How  much  did  I  buy  ? 

3.  A  boy  caught  a  fish  that  weighed  f  of  a  pound,  and 
his  sister  caught  one  weighing  |  of  a  pound.     How  much 
did  both  weigh  ? 

4.  What  is  the  sum  of  $  },  $  |,  and 


5.  A  boy  worked  |  of  a  day  for  A,  and  f  of  a  day  for 
B.     How  long  did  he  work  for  both  ? 

6.  A  man  planted  f  of  an  acre  with  potatoes,  and  f  of 
an  acre  with  corn.    How  much  land  was  planted  with  both  ? 

7.  A  grocer  sold  11  dozen  eggs  to  one  person,  f  dozen 
to  another,  and  1J  dozen  to  another.     How  many  did  he 
sell  to  all  ? 

8.  A  man  sold  three  lots,  the  first  containing  -§-  of  an 
acre,  the  second  -fa  of  an  acre,  and  the  third  1  acre.     How 
much  land  did  he  sell  ? 

9.  A  girl  paid  $  If  for  a  sled,  $  \  for  a  book,  and  $  1J 
for  a  doll.     How  much  did  her  purchases  cost  ? 

10.  James  had  $3J-,  Henry  had  f  4J,  and  Samuel  had 
$  3|.    How  much  had  they  all  ? 

11.  A  dressmaker  deposited  in  a  savings-bank  $3J  at 
one  time,  $  4f  at  another,  and  $  3£  at  another.     What  was 
the  sum  of  the  deposits  ? 


ADDITION.  HI 

12.  A  fruiterer  sold  Mr.  A  3f  dozen  bananas,  Mr.  B  2| 
dozen,  Mr.  C  4^-  dozen.     How  many  dozen  did  he  sell  them 
all? 

13.  A  carpenter  worked  4J  days  one  week,  3|-  days  the 
next,  and  5^-  days  the  next.     How  many  days  did  he  work 
in  the  three  weeks  ? 

14.  The  expenses  of  a  party  were  $  3|-  for  railroad  tickets, 
$  2f  for  carriages,  $  5^  for  provisions.     How  much  were 
the  expenses  ? 

15.  What  must  be  done  to  dissimilar  fractions  before 
they  can  be  added  ? 

What  is  the  sum  of  the  following : 

16.  f,  £,  and  f.         21.  |,  ^,  and  f.  26.  1J,  2J,  and  2|. 

17.  |,  i,  and  f.         22.  £,  |,  and  f  27.  l£,  2|,  and  3|. 

18.  i,  i,  and  f         23.  J,  ^  and  |.  28.  2J,  3£,  and  4^. 

19.  |,  A>  and  f       24.  ^,  f,  and  |.  29.  3J,  4|,  and  5|. 

20.  f,  |,  and  f          25.  £,  ^  and  f  30.  3fc  2^,  and  3f . 

150.  PRINCIPLE.  —  Only  similar  fractions  can  be  added. 

WRITTEN    EXERCISES. 

151.  1.    What  is  the  sum  of  f,  f,  and  ^? 

EXPLANATION.  —  Since    the 

i  +  t  +  A  =  fJ  +  it  +  It  =  tt     fractions  are  dissimilar,  they 
77.  __  1-3Z.  must  be   changed    to    similar 

fractions  before  adding. 

The  least  common  denominator  of  the  given  fractions  is  40,  and 
f  =  ft  5  I  =  1 1 ;  and  &  =  |f.     Hence  the  sum  is  fa  or  lf£. 


112  FRACTIONS. 

2.   What  is  the  sum  of  3J,  5£,  and  4J. 

3^  =  3^-  EXPLANATION.  —  Since  the  numbers  are  composed  of 
57  _  521  both  integers  and  fractions,  they  may  be  added  sepa- 
rately  and  their  sums  united.  Thus,  the  sum  of  the 
fractions  is  f  i,  or  l/f  ;  the  sum  of  the  integers  12  ;  and 
the  sum  of  both 


RULE.  —  Reduce  the  given  fractions  to  similar  fractions,  add 
their  numerators,  and  write  the  sum  over  the  common  denom- 
inator. 

When  there  are  mixed  numbers,  or  integers,  add  the  frac- 
tions and  integers  separately,  and  then  add  the  results. 

If  the  sum  is  an  improper  fraction,  reduce  it  to  an  integral  or  mixed 
number. 

Find  the  sum  of  the  following  : 

s.  |,  |,  I,  ft-        I*-  f  f>  A,  A-      25.  ft,  H,  M>  H- 

4-  i  I.  f>  A-  I5-  A,  A-  A.  A-     26.  f,  I,  Jf,  If. 

5-  -f.  A.  A-  A-    16-  A.  A>  M.  H-   27.  |,  ^,  «,  A- 

6-  !>  I,  A,  f-  17-  I,  A,  A>  A-       28.  i|,  if,  A,  A- 

1-  f.  f  A  f          18-  f.  1.  1.  f  29.  6f,  7|, 

s-  f  f,  I,  A        is-  A,  A.  tt  A-   so-  5f.  8f. 
»•  I.  f  A.  f-        20.  |,  A,  A,  A-     si-  7f>  6A 

10.   t  A.  A.  I-         21.  A>  I,  I,  «•         32.  8|,  9^, 

11-  f>  A-  A.  I-      22.  i,  A,  «,  H.     33.  7|,  SA, 

12-  f,  A,  A,  I-         23.  ft,  «,  If,  «.     34.  8f  6if 

13.   I,  I,  f,  A-  24.   Jj,  Jf,  ft,  A-     35.  9f,  8A>  6ff. 

36.  15},  16ft,  18ft.  39.  41#,  23^,  36if. 

37.  22ft,  18H,  19tf          40.  82JI,  18H>  45|. 

38.  35H,  26H,  84}|.          41.  43i|,  19ft,  21H- 


ADDITION.  113 

42.  What  is  the  sum  of  126f  pounds,  92f  pounds,  and 
206^-  pounds  ? 

43.  What  is  the  sum  of  1250  bushels,  720^-  bushels,  and 
640^  bushels  ? 

44.  I  bought  four  pieces  of  cloth  containing  32£,  38J, 
40-J,  and  45f  yards,  respectively.     How  many  yards  did  I 
buy  in  all  ? 

45.  Five  men  weigh,  respectively,  156J,  160|,  165^,  162^-, 
and  168f  pounds.     What  is  their  entire  weight  ? 

46.  John  has  $  1TV;  James,  $5f ;  Charles,  $17fJ;  and 
Edward,  $  12^.     How  much  have  they  together  ? 

47.  A  farmer  received  $17f  for  oats,  $28^  for  corn, 
$  76f  for  wheat,  and  $  150^  for  a  horse.     How  much  did 
he  receive  for  all  ? 

48.  Sarah  is  11^  years  old ;  Mary  is  3^-  years  older  than 
Sarah ;  and  Henry  is  5^-  years  older  than  Mary.     How  old 
is  Henry? 

49.  From  A  to  B  is  18f  miles,  from  B  to  C  20  miles, 
from  C  to  D  81f  miles,  from  D  to  E  37^  miles.     What  is 
the  distance  from  A  to  E  ? 

50.  A  merchant  sold  3f  yards  of  cloth  for  $  16|,  5£  yards 
for  $  24J,  and  8|  yards  for  $  28|.     How  many  yards  did  he 
sell,  and  how  much  money  did  he  receive  ? 

51.  A  has  13 J  acres  of  land,  B  has  17f  acres  more  than 
A,  C  has  as  much  as  both  A  and  B.     How  many  acres  has  B? 
how  many  has  C,  and  how  many  have  they  all  together  ? 

52.  A  man  has  three  fields  containing,  respectively,  16f 
acres,  18^f  acres,  and  15^J  acres.     How  many  acres  are 
there  in  the  three  fields  ? 

53.  Mr.  B  walked  23|  miles  on  Monday,  25-^  miles  on 
Tuesday,  27|f  miles  on  Wednesday,  and  29^  miles  on 
Thursday.     How  far  did  he  walk  in  all? 

8TA.ND.    AR. 8 


114  FRACTIONS. 


SUBTRACTION. 


152.    1.    Mary  had  $  -fa  and  spent  $  ^.     How  much  had 
she  left  ? 

2.  From  a  lot  containing  -J  of  an  acre  ^  of  an  acre  was 
sold.     How  large  a  lot  was  lef  t  ?     -J-  -  -|  =  ?     |  -  f  =  ? 

3.  A  boy  paid  $-J  for  his  skates,  but  sold  them  for  $^- 
less  than  he  paid  for  them.     What  did  he  get  for  them  ? 


4.  If  I  have  ^^  and  spend  $  -£$  how  much  will  I  have 
•left? 

5  .  A  girl  paid  $  |-  for  a  grammar  and  $  |-  for  a  geog- 
raphy. How  much  more  did  she  pay  for  the  geography 
than  for  the  grammar  ? 

6.  A  lad  hoed  -J  of  a  field  of  corn.     If  he  hoed  f  of  the 
field  in  the  forenoon,  how  much  did  he  do  in  the  .afternoon  ? 

7.  What  must  be  done  to  dissimilar  fractions  before 
they  can  be  subtracted  ? 

Find  the  value  of  : 


8. 

t- 

f 

15. 

f 

— 

i- 

22. 

2 

-  f- 

9. 

A- 

i 

16. 

A- 

i 
!• 

23. 

3 

-if 

10. 

|- 

2- 

17. 

A 

— 

fr 

24. 

2} 

-1!- 

11. 

A- 

f 

18. 

A 

— 

i- 

25. 

2i 

-H- 

12. 

8 

¥  ~" 

I' 

19. 

« 

— 

f 

26. 

^i 

-H- 

13. 

A- 

i« 

20. 

H 

— 

f. 

27. 

2* 

-if 

14. 

A- 

i- 

21. 

19  _     1 

sir      sir* 

28. 

2^ 

-if 

153.   PRINCIPLE.  —  Only    similar  fractions   can   be   sub' 
traded. 


SUBTRACTION.  115 

WRITTEN   EXERCISES. 
154.     1.   From  ^  subtract  f. 

9  3_36_33—  3  EXPLANATION.  —  Since  the  fractions  are 
11  ~~"4— ¥¥~~¥T--'4~4~  not  similar  they  must  be  made  similar 
before  subtracting.  The  least  common  denominator  of  the  given  frac- 
tions is  44.  T9r  =  £fandf  =  ff.  i|-if  =  £. 

2.    From  4J  subtract  2f. 

M^  __  .  8          EXPLANATION.  —  Since  the  numbers  are  composed  of 
24     integers  and  fractions,  the  integers  and  the  fractions  may 
2-g-  =  2-2-4-     ke  subtracted  separately. 

^  17         The  fractions  must  be  first  reduced  to  similar  fractions. 

»•     It  is  evident  that  |f  cannot  be  subtracted  from  ^8?,  hence 

1  or  f|  is  taken  from  4  and  united  with  the  ^  making  f  f .     ££  from 

f  f  leaves  ||,  and  2  from  3  (the  number  left  after  1  has  been  united 

with  the  fraction  ^8¥)  leaves  1.    Hence  the  remainder  is  1£|. 

E.ULE.  —  Reduce  the  fractions  to  similar  fractions.  Find 
the  difference  between  the  numerators  and  write  it  over  the 
common  denominator. 

When  there  are  mixed  numbers  or  integers,  subtract  the  frac- 
tions and  the  integers  separately. 

Mixed  numbers  may  be  reduced  to  improper  fractions  and  subtracted 
according  to  the  first  part  of  the  rule. 


19.  ff-f. 

on  1  9  __    5 

*v«  -g-g"        TB"' 

21  2  7  __    4 

22.  «--fr 

23.  H-A- 

24.  M-ii- 

25.  H-A* 

26.  41  — -14-. 


Find 

the  value  of  : 

3. 

i-f 

11. 

•g~u"~T9~' 

4. 

f-A- 

12. 

2T-A- 

5. 

A-*- 

13. 

tf-A- 

6. 

A-iV 

14. 

1H>        T5"* 

7. 

A-A- 

15. 

16         12 
4T        "3~S"" 

8. 

A—A- 

16. 

fi-A- 

9. 

1  3          18 
T5  ~~  TB"* 

17. 

~3~3~        30"* 

10. 

28  _  14 
36          26' 

18. 

H-A 

116  FRACTIONS. 

27.  ff-|f  32.      ff-ff.  37.  7f    -2jf. 

28.  ff  —  Jf.  33.  5£    —  3|.  38.  8^-5£f 

29.  ^f  — £$.  34.  6£    -4^.  39.  9      -4f. 

30.  |£_  4|.  35.  9      —  3^5,  40.  8|    — 
31-  if  — ft-  36-  8i    —  4f  41.  9A  — 

Find  the  value  of  : 

42.  £+    |-    fc  +  A*      50.  3£    +  2£    +  3^    -f-3^-  +  6f 

43.  ^-T%+    f-    f       51.   5f    +6^    -3^-    H  +  7-^. 

44-  1+    i-A  +  -jfr      52-   8i    -3f   +2*    ~2i    +5- 

45-  |+    -J-A-TV      53.   9^  +  3^-3^-lf   -2|. 

46-  A  +  A-A-rV      54.  7f   +2|    -3^-4|   +3f. 
47.    ^-+    |-    |-¥5¥.      55.  5      +6|    —  3f   —  2|   +1^-. 
48-    A—    i+    f  —    -f-      56-   7T\  — 3-f    +8H  —  5f   +9. 

58.  A  piece  of  flannel  containing  25£  yards  shrank  1| 
yards  in  dyeing.     How  much  did  the  cloth  then  measure  ? 

59.  From  a  lot  containing  |-§-  of  an  acre  of  land,  I  sold 
•A-  of  an  acre  to  one  man,  and  I-  of  an  acre  to  another.    How 

/  O  '  o 

much  land  had  I  left  ? 

60.  If  19^-  yards  are  cut  from  a  piece  of  cloth  contain- 
ing 42|-J  yards,  how  many  yards  will  be  left  ? 

61.  A  boy  gave  ISf  cents  for  a  slate,  62^-  cents  for  a 
book,  and  37^  cents  for  some  paper.     How  much  change 
should  he  receive  if  he  gave  in  payment  a  two-dollar  bill  ? 

62.  If  6  is  added  to  each  term  of  -J,  is  the  value  of  the 
fraction  increased  or  diminished,  and  how  much  ? 


MULTIPLICATION.  117 

MULTIPLICATION. 
155.    1.    How  much  is  £  of  1  of  a  yard  ?     £  of  J  of  a  yard  ? 

2.  How  much,  is  ^  of  ^-  of  a  yard  ?     J  of  ^  of  a  yard  ? 

3.  How  much  is  -^  of  J  of  an  orange  ?     -^  of  ^  of  an 
orange  ? 

4.  How  much  is  -|-  of  ^-  of  an  acre  ?     ^  of  -J  of  an  acre  ? 

5.  Since  ^  of  ^  of  an  acre  is  ^  of  an  acre,  what  part  of 
an  acre  is  J-  of  -j-  of  an  acre  ?     f  of  1  of  an  acre  ?     •£  of  £  of 
an  acre  ? 

6.  How  much  is  J  of  |-  of  a  foot  ?     J  of  J  of  a  foot  ? 

7.  Since  ^-  of  ^  of  a  foot  is  -^  of  a  foot,  what  part  of  a 
foot  is  f  of  i  of  a  foot  ?     f  of  1  of  a  foot  ?     $  of  £  of  a 
foot  ?     f  of  i  of  a  foot  ? 

8.  How  much  is  f  of  1  ?     i  of  f  ?     f  of  1  ?     j  of  |  ? 

9.  How  much  is  £  of  |  ?     J  of  |  ?     £  of  |  ?     £  of  |  ? 

10.  How  much  is  £  of  f?     J  of  f  ?     £  of  f  ?     |  of  f  ? 

11.  How  much  is  |  of  f ?     foff?     foff?     -&of$? 

12.  A  cistern  was  f  full,  but  f  of  the  water  was  drawn 
out.     What  part  of  the  amount  the  cistern  would  hold  was 
drawn  out  ? 

13.  Mr.  Ames,  who  owned  a  lot  containing  f  of  an  acre, 
sold  ^  of  it.     What  part  of  an  acre  did  he  sell  ? 

14.  If  he  had  sold  ^j-  of  it,  what  part  of  an  acre  would 
he  have  sold  ?  ^  of  f  =  ? 

15.  A  man's  farm  was  such  that  f  of  it  only  was  tilled. 
He  sold  £  of  that  part.     What  part  of  the  farm  did  he  sell  ? 


118  FRACTIONS. 

16.  A  girl  who  had  $  f  spent  f  of  it  for  candy.     What 
part  of  a  dollar  did  she  spend  ?     £  x  f  =  ?     fxf=? 

17.  A  yard  of  crape  costs  $  f .     What  will  f  of  a  yard 
cost?     |of|=?     fof£=?     £off=? 

18.  A  man  who  owned  |-  of  a  mill  sold  J  of  his  share. 
What  part  of  the  mill  did  he  sell  ?     f  of  f  =  ? 

19.  A  fruit  seller  had  f  of  a  dozen  cocoanuts  and  sold  -| 
of  them  ?     What  part  of  a  dozen  did  he  sell  ? 

20.  A  train  ran  %  of  the  distance  between  two  places  in 
an  hour.     What  part  of  the  distance  did  it  run  in  $  of  an 
hour? 

21.  Two  boys  counting  their  money  found  that  one  had 
$  f  and  the  other  had  J  as  much.     What  part  of  a  dollar 
had  each  ? 

WRITTEN   EXERCISES. 
156.    1.    Find  f  of  f,  or  multiply  £  by  f. 

3     P  4  _^  X3__3_         EXPLANATION. — To  multiply  f  by  f  is  to 
8        5~~5      8~10     find  f  of  f,  or  3  times  1  of  f    |of|  =  ^,and 
2  f  of*  =  &.<«&• 

RULE.  — Reduce  all  integers  and  mixed  numbers  to  improper 
fractions. 

Find  the  product  of  the  numerators  for  the  numerator  of  the 
product,  and  of  the  denominators,  for  its  denominator. 

1.  When  possible  use  cancellation. 

2.  The  word  o/,  between  fractions,  is  equivalent  to  the  sign  of  mul- 
tiplication.   Such  expressions  are  sometimes  called  compound  fractions. 
Thus,  |  of  -J  is  equal  to  f  x  •£. 

3.  Integers  may  be  expressed,  in  the  form  of  fractions,  by  writing 
1  as  a  denominator.     Thus,  4  may  be  written  as  f . 

Find  the  product  of : 

2.      |  X  f  4.   £2  X  |.  6.  ff  X  ^. 

3-   A  X  f  5.  If  X  f  7.   ff  X  A- 


MULTIPLICATION.  119 

8.  f|  X  H-  15-   «  X  ^  X  f        22.   2}  X  4J  X  £ 

9.  ||  X  H-  18-   H  X  H  X  f        23.   ||  X    f  X  6f. 

10.  fi  X  If-  17.  $}  X  V  X  f  24.   if  X    f  X  26}. 

11.  if  X  A  X  TV  18.  f|  X  H  X  |.  25.  ||  X  H  X  8. 

12.  ||  X  A  X  f  19.  J|  X    f  X  |.  26.   JJ  X  3}  X  •&• 

13.  ||  X  A  X  TV  20.  i  X    f  X  2i  27.  |f  X  ^  X  4f 
14-  tt  X  H  X  |.  21.  |  X  7}  X  |.  28.   ||  X  |  X  9. 

29.    Multiply  2|  by  10. 

2|  EXPLANATION.  — In  examples  like  this,  it  is  best  to  multiply 

-JQ  the  fraction  and  integer  separately  and  to  add  the  results.    Thus, 

—  10  times  2  are  20.     10  times  f  are  3/,  or  3f ,  or  3|.     This  added 

^  to  20,  gives  the  entire  product,  23f.     Or  their  product  may  be 

3J  found  by  the  general  rule.     Thus, 

23f  2f  =  Y-,  and  -y>-  x  -V0-  =  if^  =  23|. 

Multiply  the  following : 

30.  12|byl5.  35.  29^  by  26.  40.  24||  by  48. 

31.  15|  by  24.  36.  41T%  by  41.  41.  25||  by  35. 

32.  24f  by  20.  37.  32^  by  39.  42.  31|f  by  33. 

33.  36f  by  18.          38.  24^-  by  32.  43.  18||  by  25. 

34.  42|  by  36.  39.  29^  by  34.  44.  39|f-  by  27. 

45.   Multiply  10  by  2|. 

10  EXPLANATION. — Multiply  by  the  integer   and  the  fraction 

03     separately,  and  add  the  results. 

— *         Thus,  2  times  10  =  20.     ^  of  10  =  \°-,  and  f  of  10  =  -\°-  or,  3|. 
This  added  to  20  gives  the  entire  product,  23|.     Or,  the  product 

3-|     may  be  found  by  the  general  rule.    Thus, 

23f  \°-  x  -V9-  =  -^  or  23|. 


120  FRACTIONS. 

Multiply : 

46.  25by3|.  54.  52  by  4f .  62.  8|    by  20. 

47.  30  by  4-j^.  55.  61  by  8f.  63.  9^-  by  25. 

48.  56  by  7f.  56.  59  by  7f.  64.  8^-  by  21. 

49.  63by8|.  57.  43  by  4f  65.  9f£  by  18. 

50.  64by7f.  58.  63  by  6£.  66.  9^  by  26. 

51.  60  by  8^.  59.  71  by  8f .  67.  7^  by  34. 

52.  72  by  9f .  60.  3|    by  24.  68.  6^-  by  27. 

53.  42by8f  61.  5-^  by  35.  69.  8|i  by  28. 

70.  9i|by26.  79.  36  x  5f    x  4|. 

71.  7^  by  38.  80.  38  x  6f    x  5f. 

72.  9if  by  84.  81.  42  x  7T77  x  4f. 

73.  9ffby52.  82.  45  x  8|    X  7^. 

74.  25x3|    x4^.  83.  47x6^x5|. 

75.  15  x  4^    x  5J.  84.  49  x  2f    x  4f 

76.  24  x  6^  x  6|.  85.  50  x  3-^  x  7f. 

77.  25  x  3f    x  6^.  86.  52  x  7T%  x  4f 

78.  40x4f    x3f.  87.  55x6^x6^. 

88.  A  coat  cost  $  12^,  and  a  hat  -§•  as  much.     What  was 
the  cost  of  the  hat  ? 

89 .  How  much  will  121  yards  of  cloth  cost,  at  $  3f  a  yard  ? 

90.  I  bought  66f  yards  of  flannel,  at  $.37-^-  per  yard. 
How  much  did  I  pay  for  it  ? 

91.  What  will  43 J  bushels  of  potatoes  cost,  at  $  .621  per 
bushel  ? 

92.  A  man  sold  7f  tons  of  hay,  at  $  15^-  per  ton.     How 
much  did  he  receive  for  it  ? 


MULTIPLICATION.  1 21 

93.  If  a  man  earns  $1}  per  day,  how  much  can  he  earn 
in  35  days  ? 

94.  James  has  $  4f ,  and  Mary  has  f  as  much.    How  much 
money  has  Mary  ? 

95.  At  f  If  each,  what  will  65  Latin  grammars  cost  ? 

96.  What  are  18J  bushels  of  apples  worth,  at  $•£•  per 
bushel  ? 

97.  If  one  load  of  hay  is  worth  $6f,  how  much  are  18 
loads  worth  ? 

98.  A  has  f  of  $  125,  and  B  has  f  as  much  as  A.     How 
much  money  has  B  ? 

99.  Mr.  B  owns  f  of  a  farm  valued  at  $  16728.     What  is 
the  value  of  his  portion  of  the  farm  ? 

100.  A  hotel  in  one  month  used  31  pounds  of  coffee,  and 
7f  times  as  much  sugar.     How  much  sugar  was  used  ? 

101.  A  merchant  bought  a  piece  of  cloth  for  $57},  but 
was  obliged  to  sell  it  for  %  of  what  it  cost  him.     How  much 
did  he  lose  ? 

102.  John  can  walk  21^-  miles  in  a  day.     How  far  can 
he  walk  in  24  days  ? 

103.  What  must  be  paid  for  17  tables,  at  $  7f  apiece  ? 

104.  A  and  B  bought  a  mowing  machine  for  $75.     A 
paid  £  of  the  cost,  and  B  -|.     How  much  did  each  pay  ? 

105.  A  wheel  in  making  one  revolution  travels  15^-  feet. 
How  far  will  it  travel  in  making  25  revolutions  ? 

106.  What  will  be  the  cost  of  f  of  a  piece  of  cloth  con- 
taining 231  yards,  at  $  3^-  per  yard  ? 

107.  If  a  ship  sails  18J-  miles  an  hour,  how  far  will  she 
sail  in  15  hours  ? 


122  FRACTIONS. 

DIVISION. 
157.    1.    How  many  times  is  £  contained  in  1  ? 

2.  Since  -^  is  contained  in  1  three  times,  what  part  of 
three  times  is  -|  contained  in  1  ? 

3.  How  many  times  is  ^  contained  in  1  ?     Since  -J-  is 
contained  in  1  five  times,  what  part  of  5  times  is  -§-  con- 
tained in  1  ?     |?     f? 

4.  Since  f  is  contained  in  1  -J  times,  how  many  times 
will  it  be  contained  in  1  ?     In  £  ?     In  £  ?     In  f  ? 

5.  How  many  times  is  \  contained  inl?     f?     |-  ?     %? 

6.  Since  ^  is  contained  in  1  -J  times,  how  many  times 
will  it  be  contained  in  %  ?     In  £  ?     In  f  ?     In  f  ? 

7.  How  many  quires  of  paper,  at  $•£•  per  quire,  can  be 
purchased  f  or  $  -^  ?     For  $  ^  ?     For  $  -&  ? 

8.  At  $-|  per  yard,  how  much  cloth  can  be  bought  for 
$|?     For$£?     For$f? 

9.  At  $f  per  pound,  how  many  pounds  of  raisins  can 
be  bought  for  $1?     For  $1?     For  $f  ? 

10.  When  rye  is  worth  $  |-  per  bushel,  how  much  can  be 
purchased  for  $  1  ?     For  $  |  ?     For  $  1£  ? 

11.  At  $-|  per  pound,  how  much  honey  can  be  bought 
for$l?     For  $2?     For$-i?     For  $3? 

12.  How  many  pictures,  worth  $  f  each,  can  be  purchased 
for  $3?     For  $6?     For  $9? 

13.  How  long  will  it  take  a  boy  to  earn  $  1J,  if  he  earns 
$  f  per  day  ?     How  long  to  earn  $  2  J  ? 

14.  If  a  man  pays  f  £  per  day  for  his  board,  for  how 
pany  days'  board  will  $  4  pay  ?     $  8  ?     $  12  ? 


DIVISION.  123 

15.  The  cost  of  coal  is  $f  per  hundred-weight.     How 
many  hundred-weight  can  be  bought  for  $  1  ?     How  many 
for  $2? 

16.  Some  diaries  are  sold  for  $f  each.     How  many  can 
be  bought  for  $  3  ?     How  many  for  $  6  ? 

WRITTEN    EXERCISES. 

158.   1.    Divide  f  by  f,  or  find  how  many  times  f  is  con- 
tained in  -|. 

4_±_3_4x7-=28    or  113  EXPLANATION.  —  \  is  contained  in 

1  seven  times,  and  f  is  contained  in 
1,  one  third  of  7  times,  or  |  times. 

Since  f  is  contained  in  1  f  times,  in  f  it  will  be  contained  f  of  ^ 
times,  or  f  f  times,  or  lif  times. 

KULE.  —  Multiply  the  dividend  by  the  divisor  inverted. 

1.  When  possible,  use  cancellation. 

2.  Integers  and  mixed  numbers  must  be  reduced  to  improper  frac- 
tions. 

Find  the  quotients  of : 

2.  f-f-f.         is.  |f -5- f         24.  2|-*-f         35.  17 -*- 3 £. 

3.  A  +  i         14'  It-tt-       25.  3J-hf         36.  18-n4fc 

4.  T\-|.         15.  HH-H,       26.  6f -*•¥•       37'  17-3i- 
6.  «  H-  f         16.  «  -H  }f       27.  8|  -h  |.         38.  20  -  3f. 

6.  |f +  f  17.  !  +  £  28.  9|^-if.  39.  f-s-10. 

7.  ^|^_6.  18.  5^.2.  29.  7f -*-!£,  40.  f-j-8. 

8.  «-Hf.  19.  6 -A-  30-  8i-2f  41-  T-1^ 

9.  if-f-f  20.  8-5-f.  31.  8|H-3^.  42.  A-12- 

10.  if-*-f         21.     9-f         32.  6f-i-3J..       43.  -^-i-18. 

11.  |4^_|.         22.     8-4-f.         33.  4|  +  2f       44.  ^-i-16. 

12.  ff  +  f.          23.      7-TV        34.   15-!-3|.        45.  ff  +  30. 


124  FRACTIONS. 

Divide : 

46.  f  x  |  by  16  -j-  f  57.  1G*  X  7  by  ff 

47.  ££-*-£  by  |  of  9.  58.  91  x  4f  by  6  x  f . 

48.  |f  X  A  by  f  of  2f  59.  12f  X  |  by  18J. 

49.  |f  x  f  by  |  of  21  60.  |  x  4f  by  f  x  3f. 

50.  «  X  H  by  f  of  3.  61.  |t  x  ft  by  if  x  ft. 

51.  If  X  |  by  li  X  f.  62.  ||  X  A  X  «  by  14|. 

52.  fj  X  fi  X  A-  by  2J.  63.  48  x  ff  x  8  by  17|. 

53.  if  X  ^  X  A  by  8|.  64.  ff  X  7-^  by  f  x  5f 

54.  «  by  |  X  |  X  A.  65.  |f  X  if  X  51  by  18f. 

55.  6^  by  &  x  |f  X  19.  66.  If  x  14f  by  2Qf 

56.  14|  X  |  by  if  x  TV  67.  if  X  H  X  3-1  by  27  XTV 

68.  Divide  f  of  f  of  f  of  3£  by  |  of  i|  of  f  of  |  of  6. 

SUGGESTION.  — Change  the  integers  and  mixed  numbers  to  improper 
fractions,  and  invert  all  factors  of  the  divisor.     Use  cancellation. 

|  of  f  of  |  of  3^  divided  by  |  of  \\  of  f  of  f  of  6 

«  f  x  f  x^  x  V  x  I  x  if  x  I  x  f  x  f  =  ft  =  1A 

69.  Divide  |  of  f  of  |  of  -f  by  f  of  f  of  -^  of  4. 

70.  Divide  -fy  of  if  of  i|  of  4J-  by  f  of  if  of  i|. 

71.  Divide  |  of  |f  of  Jf  of  7|  by  i|  of  i|  of  141. 

72.  Divide  A  of  ft  of  #  of  if  by  if  of  |f  of  ^. 

73.  Divide  ^  of  ^  of  ||  of  7f  by  ^  of  ^  of  Jf 

74.  Divide  J|  of  14  of  |J  of  fi  by  if  of  ||  of  f  of  8|. 

75.  Divide  |  of  ||  of  -f  of  5f  by  ^-  of  J|  of  ^-  of  4f 

76.  Divide  if  of  If  of  -H  of  71  by  ||  of  f  of  if  of  ^  of 


DIVISION  125 

77.  Divide  16 f  by  5. 

5)16|.  EXPLANATION.  —  The  division  may  be  performed  in  the 

o  7        ordinary  way,  but  it  is  often  more  convenient  to  divide  as 

2?       follows : 

5  is  contained  in  16|,  3  times  with  a  remainder  of  If  or  | ;  and  J 
divided  by  5  equals  ^V    Therefore  the  quotient  is  3^. 

Divide : 

78.  28|  by  6.  83.  42|  by  8.  88.  24|  by  7. 

79.  35}  by  8.  84.  33f  by  6.  89.  32£  by  5. 

80.  61fby7.  85.  36|  by  9.  90.  40f  by  6. 

81.  52 J  by  5.  86.  28f  by  6.  91.  261  by  5. 

82.  411  by  8.  87.  30^  by  7.  92.  33f  by  4. 

93.  Divide  27  by  3J. 

27  EXPLANATION.  —  It    is    frequently  convenient    to 
4           reduce  the  numbers  to  equivalent  fractions  having 
the  same  denominator,  and  then  to  divide  the  nume- 


108  rator  of  the  dividend  by  the  numerator  of  the  divisor. 


Thus,       3^  =  13  fourths,  and  27  =  108  fourths. 
Then       108  -4- 13  =  8T4T,  the  quotient. 


13 


94.  35  by  if  98.  54by5£.  102.  61    by 

95.  42  by  3|.  99.  36  by  4f.  103.  24f  by 

96.  37  by  6f  100.  48  by  7|.  104.  18|  by 

97.  29by4|.  101.  35  by  6f  105.  13J  by  4|. 

106.  If  a  yard  of  silk  costs  $2£,  how  many  yards  can  be 
bought  for  $15£? 

107.  At  $  6f  per  barrel,  how  many  barrels  of  flour  can  be 
bought  f  or  $74^? 

108.  If  a  man  can  walk  3|  miles  an  hour,  in  how  many 
hours  can  he  walk  30f  miles  ? 


126  FRACTIONS. 

109.  Mr.  Hill  can  cut  3f  cords  of  wood  in  one  day.     How 
many  days  will  lie  require  to  cut  38-f$  cords  ? 

110.  I  bought  18  turkeys  for  $  32f.    What  was  the  aver- 
age price  of  each  ? 

111.  A   man  divided  $16  among   some   poor  children, 
giving  to  each  $  If.     How  many  children  received  a  share  ? 

112.  A  shoe  dealer  paid  $  60  for  a  case  of  overshoes,  at  $  f 
a  pair.     How  many  pairs  of  shoes  were  there  in  the  case  ? 

113.  Mr.  Long  earns  $  26f  in  a  week,  or  six  working  days. 
How  much  does  he  earn  per  day  ? 

114.  How  many  yards  of  cloth  can  be  bought  for  $  65,  at 

$  !  a  yard  ? 

115.  What  is  the  average  weight  of  12  men  whose  united 
weight  is  1768f  pounds  ? 

116.  A  farmer  raised  536f  bushels  of  wheat  from  a  field 
containing  21  acres.     What  was  the  average  yield  per  acre  ? 

117.  How  many  steps  will  it  take  to  walk  a  mile,  or  5280 
feet,  each  step  being  2-|  feet  in  length  ? 

118.  At  $33J  per  acre,  how  many  acres  of  land  can  be 
purchased  for  $  6750|  ? 

119.  I  paid  $  15f  for  25  bushels  of  potatoes.     What  did 
I  pay  per  bushel  ? 

120.  In  a  field  containing  3f  acres  there  were  raised  87 
bushels  of  grain.     What  was  the  yield  per  acre  ? 

121.  Four  men  in  partnership  gain  $4327J.     If  they 
share  equally,  what  will  be  each  one's  share  of  the  gain  ? 

122.  What  is  the  price  of  hay,  when  5|  tons  are  worth 


123.    I  paid  $36^  for  6£  cords  of  wood.     What  was  the 
price  per  cord  ? 


FRACTIONAL  FORMS.  127 


FRACTIONAL  FORMS. 

159.  Expressions  of  unexecuted  division  of  fractions  are 
often  written  in  the  form  of  a  fraction.  Such  expressions 
are  usually  termed  Complex  Fractions. 

Thus,  f-r-5,  written  |,  and  6|-j-f,  written  -^,  are  complex  fractions. 
1.   Find  the  value  of  -f- 

1) 

EXPLANATION.  — The  expression  means  3|  -s-  f,  and  is  solved  like 
other  examples  in  division  of  fractions. 

Reduce  to  simple  fractions  : 


2. 

If             7    6 

r       '  f 

12.    9. 

1?'  fof2i 

3. 

it             8.  21 

is.  a 

is.  I°f2i 

1                      * 

2 

|-  of  |^ 

6* 

4. 

21                               85 

9     JL 

ft                   12" 

J-TC  .                      • 

1Q     1  of  3f 

'    |of2f 

~2 

5. 

fi.            10.   —  . 

15.  a 

20.    ^Of5^ 

T\                          A 

3 

¥of  5i 

6.    M.  11.    rr.  16.    ±^5.  21. 


128  FRACTIONS. 

FRACTIONAL  RELATION  OP  NUMBERS. 
160.   To  find  what  part  one  number  is  of  another. 
1.    What  part  of  8  is  2£? 

SOLUTION.  —  1  is  |  of  8  ;  hence,  21,  is  2£  times  1  of  8,  or  -^  of  8,  or 
A  of  8. 

What  part  of 

2.  8  is  7  ?         6.  13  is  11  ?  10.  10  is  3£  ?  14.  18  is  f  ? 

3.  9is3?         7.  15is8?  11.  12  is  3J?  15.  15  is  f? 

4.  18  is  5  ?       8.  14  is  7  ?  12.  15  is  4£  ?  16.  21  is  f  ? 

5.  21is6?       9.  24isl3?  18.  13  is  2£?  17.  35  is  f? 

18.  What  part  of  4J-  is  2  ? 

SOLUTION.  —  4|  =  f ,  and  2  =  f .    f  is  f  of  f .    Therefore  2  is  f  of  4£. 

What  part  of 

19.  5J  is  3  ?       23.  4-f  is  2  ?  27.  8|  is  6  ?  31.  f  is  12  ? 

20.  5|  is  5  ?       24.  5J  is  3  ?  28.  7|  is  5  ?  32.  f  is  8  ? 

21.  7£is2?       25.  3fis2?  29.  8fis6?  33.  f  is  10  ? 

22.  6^- is  4?       26.  6f  is  5  ?  30.  9£  is  8  ?  34.  $  is  9? 

35.    What  part  of  4J  is  2J  ? 

SOLUTION.  —  4|  =  -2/-  and  2£  =  -^.    -^*  is  ^  of  -^.    Therefore  2^  is 

If  of  4|. 

What  part  of 

36.  21  is  3J  ?     40.  3^  is  2f  ?  44.  6|  is  3J  ?  48.  f  is  |  ? 

37.  3Jis4i?     41.  3f  islj?  45.  5f  is  4^.?  49.  f  isf? 

38.  4|  is  3J-  ?     42.  5^  is  2f  ?  46.  3f  is  2f  ?  50.  -^  is  f  ? 

39.  6£  is  3J  ?     43.  8f  is  3|  ?  47.  4|  is  3J  ?  51.  f  is  f  ? 


FRACTIONAL  RELATION  OF  NUMBERS.         129 

161.  A  number  and  its  relation  to  another  number  given, 
to  find  the  other  number. 

1.    120  is  f  of  what  number  ? 

SOLUTION.  —  Since  120  is  |  of  a  number,  |  of  the  number  is  £  of 
120,  or  30 ;  and  since  30  is  ^  of  the  number,  the  number  must  be  5 
times  30,  or  150.  Hence,  120  is  f  of  150. 

Find  the  number  of  which 

2.  160  is  f.       7.  240  is  f     12.  380  is  |.  17.  510  is  |£. 

3.  180  is  f.       8.  252  is  f.     13.  392  is  f.  18.  756  is  |£. 

4.  176  is  f.       9.  275  is  |.     14.  415  is  ^-.  19.  589  is  -JJ. 

5.  180  is  |.     10.  320  is  f.     15.  474  is  T%.  20.  609  is  -§-£. 

6.  192  is  f.     11.  364  is  £.     16.  497  is  •&.  21.  625  is  ff. 

22.  A  man  spent  $450,  which  was  -|  of  his  money.     How 
much  money  had  he  ? 

23.  Mr.  B  bought  a  house  for  $1260.     The  house  cost 
him  f  as  much  as  his  store.     What  did  he  pay   for  the 
store  ? 

24.  A  drover  paid  $2914  for  a  lot  of  sheep,  which  is  -^ 
of  what  he  sold  them  for.     How  much  did  he  receive  for 
them  ? 

25.  C  bought  a  cow  for  $  31.50,  which  was  ^-  of  what  he 
paid  for  a  horse.     What  was  the  cost  of  the  horse  ? 

26.  A  lady  bought  a  dress  for  $  21.65  and  found  that  she 
had  spent  -§-  of  her  money.     How  much  money  had  she  ? 

27.  D  has  a  library  which  contains  1264  books.     His 
library  contains  -£  as  many  books  as  E's.     How  many  books 
are  there  in  E's  library  ? 

STAND.    AR. 9 


130  FRACTIONS. 

28.  A  farmer  raised  672  bushels  of  wheat,  which  was  ^- 
of  the  number  of  bushels  of  corn  he  raised.     How  much 
corn  did  he  raise  ? 

29.  Mr.  H  built  a  barn  which  cost  him  $2513.      The 
bar  a  cost  him  -^  as  much  as  his  house.     What  was  the  cost 
of  the  house  ? 

30.  A  man  burned  1560  bushels  of  lime  in  September, 
which  was  -f-  of  the  number  of  bushels  burned  in  October. 
How  much  lime  did  he  burn  in  October  ? 

31.  A  merchant  sold  on  Monday  goods   amounting  to 
$  376.70,  which  was  -|  of  the  amount  received  for  goods  on 
Tuesday.     What  was   the  amount  received  for  Tuesday's 
sale? 

32.  Mr.  K  deposited  in  one  bank  $  1936,  which  was  ^J- 
of  the  money  which  he  deposited  in  another  bank.     How 
much  did  he  deposit  in  the  latter  bank  ? 

33.  A  man  left  his  son  $19,000,  which  was  -f  the  value 
of  his  estate.     What  was  the  value  of  his  estate  ? 

34.  3300  feet  are  |-  of  a  mile  ?    How  many  feet  are  there 
in  a  mile  ? 

35.  A  man  sold  his  house  for  $36,000,  which  was  -^  of 
it?  cost.     How  much  did  the  house  cost  ? 

36.  Mr.  Black  spends  $48  a  week,  which  is  f  of  his 
weekly  income.     Mr.  Cutler  spends  $  60  a  week,  which  is 
•f  of  his  weekly  income.     Which  of  them  has  the  greater 
income,  and  how  much  greater  ? 

37.  A  man  has  debts  amounting  to  $2193.     This  amount 
is  equal  to  ±~£  of  his  resources.     How  much  money  is  he 
worth  ? 

38.  A  man  expended  -J-g-  of  his  fortune  in  buying  a  farm 
for  $  4826.     What  was  his  fortune  ? 


EXE 


REVIEW  EXERCISES.  131 

REVIEW  EXERCISES. 
ORAL    EXERCISES. 

162.   1.    A  boy  having  $  £  earned  $f.     What  part  of  a 
dollar  had  he  then  ? 

2.  Mr.  A,  having  3f  acres  of  land,  bought  6f  acres. 
How  much  land  did  he  then  have  ?     He  then  sold  -fa  of  an 
acre.     How  much  had  he  leftf? 

3.  A  man  owning  f  of  a  ship  sold  -§-  of  his  share.     What 
part  of  the  ship  does  he  still  own  ? 

4.  Mr.  A  buying  a  farm  paid  -1-  cash,  \  the  second  year, 
and  -|  the  next  year.     How  much  more  must  he  pay  to  own 
the  farm  ? 

5.  A  boy  bought  a  watch  for  $27,  which  was  4^  times 
the  cost  of  the  chain.     What  did  the  chain  cost  ? 

6.  Mr.  K  is  45  years  old  and  his  wife  is  -|  of  his  age. 
How  old  is  his  wife  ? 

7.  A  horse  costs  $160,  and  f  of  its  cost  is  3  times  the 
cost  of  a  cow.     What  is  the  cost  of  the  cow  ? 

8.  If  to  the  height  of  a  certain  tree  in  California  you 
add  J  of  its  height,  the  sum  will  equal  266  feet.      How 
high  is  the  tree  ? 

9.  How  many  pounds  of  coffee,  at  ^  of  a  dollar  a  pound, 
can  be  bought  for  $  6  ? 

10.  A  man  bought  turkeys  at  If  dollars  apiece.     How 
many  did  he  get  for  $  21  ?    How  many  for  $  28?    For  $  35  ? 
For  $56? 

11.  If  a  man  can  plow  ^  of  a  field  containing  5  acres  iu 
a  day,  how  much  can  he  plow  in  f  of  a  day  ? 


132  FRACTIONS. 

12.  How  much  will  5  carpenters  earn  in  4  days  if  they 
each  receive  $  2^-  per  day  ? 

13.  If  there  are  2f  bushels  of  apples  in  a  barrel,  how 
many  barrels  will  contain  22  bushels  ? 

14.  A  can  do  a  certain  piece  of  work  in  3  days,  and  B  can 
do  it  in  4  days.     How  long  will  it  take  them  to  do  it  work- 
ing together  ? 

SOLUTION.  —  Since  A  can  do  the  work  in  3  days,  he  can  do  i  of  it 
in  1  day  ;  and  since  B  can  do  the  work  in  4  days,  he  can  do  £  of  it  in 
1  day.  Both  together  can  therefore  do  the  sum  of  i  and  £  or  ^  of  it 
in  1  day.  If  both  did  T\  of  the  work  in  a  day,  they  could  do  the  whole 
in  12  days,  but  inasmuch  as  they  do  T^  of  the  work  in  1  day,  they  can 
do  the  whole  in  }  of  12  days,  or  If  days. 

15.  A  can  do  a  certain  piece  of  work  in  2  days,  B  in  3 
days,  and  C  in  4  days.     How  long  will  it  take  them  to  do 
it  working  together  ? 

16.  Charles  lost  ^  of  his  marbles,  and  then  had  21  left. 
How  many  marbles  had  he  at  first  ?     How  many  marbles 
did  he  lose  ? 

17.  A  grocer  sold  10£  pounds  of  butter  for  $  4.20.     What 
was  the  price  per  pound  ? 

18.  After  selling  4J  acres  to  A,  and  5f  acres  to  B,  I  find 
that  I  have  f  of  my  land  left.     How  much  had  I  at  first  ? 

19.  If  8  men  eat  4  loaves  of  bread  in  one  day,  how  many 
loaves  will  5  men  eat  in  3  days  ? 

20.  A  man  is  42  years  of  age,  and  fy  of  his  age  is  \  of  his 
wife's  age.     How  old  is  his  wife  ? 

21.  Mr.  H  bought  a  cow  for  $35  and  sold  it  for  -f-  of 
what  it  cost.     How  much  did  he  lose  ? 

22.  A  farmer  sold  60  sheep,  which  were  |-  of  what  re- 
mained.    How  many  had  he  at  first? 


REVIEW  EXERCISES.  133 

23.  A  man  sold  £  of  his  farm  for  $  1500.     At  this  rate 
what  was  the  value  of  the  farm  ? 

24.  A  man  sold  f  of  his  farm  and  had  30  acres  left.     How 
many  acres  had  he  at  first  ? 

25.  Edward  gave  $.60  for  a  book  and  a  slate,  and  the 
slate  cost  -|  as  much  as  the  book.     What  did  each  cost  ? 

SOLUTION.  —  The  book  cost  a  certain  sum  and  the  slate  cost  f  as 
much  ;  hence  both  must  have  cost  If  or  ty  times  the  cost  of  the  book. 
Then,  since  -^  of  the  cost  of  the  book  was  60  cents,  \  of  the  cost  was 
Y1^  of  60  cents,  or  6  cents.  Since  \  of  the  cost  of  the  book  was  6  cents, 
the  entire  cost  of  the  book  was  7  times  6  cents,  or  42  cents.  Hence  the 
book  cost  42  cents  and  the  slate  60  cents  — 42  cents,  o*-18  cents. 

26.  A  horse  cost  $125,  and  f  of  the  cost  of  the  horse 
is  4  times  the  cost  of  the  harness.     What  did  the  harness 
cost? 

27.  If  a  rod  4  feet  long  casts  a  shadow  6|-  feet  long,  what 
is  the  length  of  the  shadow  that  a  rod  12  feet  long  will  cast 
at  the  same  time  of  day  ? 

28.  A  can  do  a  piece  of  work  in  6  days,  and  B  can  do  it 
in  10  days.     In  what  time  can  both  do  it  ? 

29.  If  f  of  a  yard  of  cloth  costs  $3f,  what  will  5  yards 
cost? 

30.  A  market  woman  bought  eggs  at  the  rate  of  4  for 
5  cents,  and  sold  them  at  the  rate  of  5  for  9  cents.     How 
much  did  she  gain  on  each  egg  ? 

31.  How  many  oranges,  at  7-J-  cents  apiece,  can  be  ex- 
changed for  60  pears,  at  2^-  cents  apiece  ? 

32.  A  fruit  grower  sold  28  bushels  of  peaches,  and  had 
£  of  his  peaches  left.     How  many  peaches  did  he  have  ? 

33.  If  the  yearly  rent  of  a  house  is  $600,  what  will 
be  the  rent  for  1\  months  at  the  same  rate  ? 


134  FRACTIONS. 

34.  If  3  boxes  of  oranges  cost  $  5-|,  how  many  boxes  can 
be  bought  for  $  17. 

35.  Henry  hoed  3f  acres  of  corn,  hoeing  ^  of  an  acre  each 
day.     How  many  days  did  it  take  him  ? 

36.  Mary  is  f  as  old  as  her  mother,  who  is  40  years  of 
age.     Her  mother  is  -^  of  the  age  of  her  grandmother.     How 
old  is  Mary,  and  how  old  is  her  grandmother  ? 

37.  f  of  a  cord  of  wood,  at  $  6  per  cord,  will  pay  for  what 
part  of  a  ton  of  coal,  at  f  8  per  ton  ? 

38.  A  sold  B  |  of  his  land,  and  then  bought  back  % 
of  what  he  had  sold.     What  part  of  the  land  did  each  then 
have? 

39.  I  raised  from  my  orchard  75  bushels  of  apples.     I 
kept  25  bushels  for  my  own  use,  gave  lOf  bushels  to  a 
friend,  and  4^-  bushels  decayed.     The  rest  I  sold.     How 
many  bushels  did  I  sell  ? 


40.  Jacob  and  Rebecca  have  each  $>  3^.     They  agree  to 
buy  a  book  costing  $  3J  as  a  Christmas  present  for  their 
mother.     How  much  has  each  left  if  they  share  the  expense 
equally  ? 

41.  Charles  lives  1-1-  miles  from  the  schoolhouse.     If  it 
takes  him  6|  minutes  to  go  ^  of  a  mile,  how  long  will  it 
take  him  to  go  to  school  ? 

42.  How  many  bushels  of  potatoes,  at  $J  a  bushel,  will 
pay  for  a  barrel  of  sugar,  at  $  1\  ? 

43.  A  pole  is  ^  in  the  mud,  \  in  the  water,  and  44  feet  in 
the  air.     How  long  is  the  pole  ? 

44.  A  fox  is  60  rods  in  advance  of  a  hound.     The  fox 
runs  62  rods  a  minute,  the  hound  66.     How  many  minutes 
will  it  take  the  hound  to  catch  the  fox  ? 


REVIEW  EXERCISES.  135 

45.  What  is  the  value  of  20  bushels  of  grain,  at  the  rate 
of  3£  bushels  for  $2? 

46.  What  is  the  value  of  f  of  an  acre  of  land,  if  £  of  an 
acre  is  worth  $  84  ? 

47.  How  much  will  6£  barrels  of  flour  cost,  if  f  of  a 
barrel  costs 


48.  If  3|  yards  of  cloth  are  worth  $  9|,  what  is  a  yard 
worth  ? 

49.  A  man  worked  llf  days,  and  after  paying  his  board 
and  other  expenses  with  $  of  his  earnings,  had  $  15  left. 
How  much  did  he  receive  a  day  ? 

50.  A  and  B  do  a  piece  of  work  in  7  days,  and  B  can  do 
\  of  the  same  in  3£  days.     How  long  will  it  take  each  to 
do  it  alone  ? 

SOLUTION.  —  Since  B  can  do  \  of  the  work  in  3£  days,  he  can  do 
the  whole  work  in  4  times  3|  days,  or  14  days,  and  -fa  of  it  in  1  day. 
Since  A  and  B  can  do  \  of  it  in  1  day,  and  B  can  do  ^  of  it  in  1  day, 
\  -  fa  or  fa  is  the  part  that  A  can  do  in  1  day.  Since  A  can  do  ^  of 
the  work  in  1  day,  he  can  do  the  whole  work  in  14  days.  Hence  each 
can  do  the  work  in  14  days. 


WRITTEN   EXERCISES. 

163.  1.  A  farm  is  divided  into  five  fields  containing 
respectively  19f  ,  28|,  30f  ,  36|,  and  39£  acres.  How  many 
acres  are  there  in  the  farm? 

2.  I  bought  a  barrel  of  flour  for  $  6f  ,  3  bushels  of  potatoes 
at  $  f  per  bushel,  and  a  ham  for  $  3|,  and  gave  the  clerk  a 
twenty-dollar  bill.     How  much  change  did  I  get  ? 

3.  If  a  man  working  8-J-  hours  a  day  can  finish  a  piece  of 
work  in  12  days,  how  many  hours  per  day  must  he  work  to 
complete  it  in  8f  days  ? 


136  FRACTIONS. 

4.  What  is  the  value  of  700  eggs,  at  25  cents  per  dozen  ? 

5.  The  quotient  is  347  and  the  divisor  26|.    What  is  the 
dividend  ? 

6.  A  sold  f  of  his  land,  and  then  had  115J  acres.     How 
much  land  had  he  before  the  sale  ? 

7.  There  are  30^  square  yards  in  a  square  rod.     How 
many  square  rods  are  there  in  1000  square  yards  ? 

8.  A  man  bequeathed  to  his  wife  $  5604,  which  was  -i-f-  of 
his  estate.     What  was  the  value  of  his  estate  ? 

9.  If  6  is  added  to  both  terms  of  the  fraction  -J,  will  the 
value  of  the  fraction  be  increased  or  diminished,  and  how 
much? 

10.  If  6  is  subtracted  from  both  terms  of  the  fraction  1, 
will  its  value  be  increased  or  diminished,  and  how  much  ? 

11.  A  merchant  bought  450  pounds  of  sugar  at  4^  cents 
a  pound,  50  pounds  of  tea  at  37^-  cents  a  pound,  and  80 
pounds  of  rice  at  8J  cents  a  pound.     What  was  the  entire 
cost? 

12.  A  grocer,  after  selling  |,  ^  -J-,  and  £  of  a  quantity  of 
sugar,  had  260  pounds  left.    How  many  pounds  had  he 
at  first  ? 

13.  A  owns  -f-  of  a  section  of  land,  B  f  of  a  section,  and 
C  ^5-  as  much  as  A  and  B  together.     What  part  of  a  section 
does  C  own  ? 

14.  Divide  f  of  £  of  ^  of  3f  by  6  x  f  of  f  of  5. 

15.  If  12  J  bushels  of  potatoes,  at  40  cents  a  bushel,  are 
given  for  a  quantity  of  molasses,  at  18f  cents  a  gallon,  how 
many  gallons  of  molasses  are  obtained  ? 

16.  From  a  barrel  of  vinegar  containing  411  gallons,  -fa 
leaked  out.     If  I  paid  $  6.75  for  it,  at  what  price  per  gallon 
must  I  sell  the  remainder  to  still  gain  $  1  on  the  vinegar  ? 


REVIEW  EXERCISES.  137 


17.  A  saleswoman  earns  f^  a  day,  and  her  expenses  are 
$  3^9.  a  week.     How  much  money  can  she  save  in  a  year,  or 
52  weeks  ? 

18.  A  stock  of  goods  was  owned  by  three  partners.     A 
owned  f  of  it,  B  f  of  it,  and  C  the  remainder.     The  goods 
were  sold  at  a  profit  of  $  4260.     What  was  each  one's  share 
of  the  gain  ? 

19.  A  man  worked  24£  days,  and  after  paying  for  his 
board  and  other  expenses  with  -^  of  his  earnings,  he  had  $  24 
remaining.     What  were  his  daily  wages  ? 

20.  If  a  miller  takes  -^  of  the  quantity  of  grain  for  grind- 
ing it,  how  many  bushels  must  a  farmer  carry  to  the  mill 
that  he  may  take  away  21  bushels  of  ground  grain  ? 

21.  I  paid  9  cents  a  pound  for  a  live  turkey  that  weighed 
16f  pounds,  and  the  waste  in  dressing  was  -?-  of  its  weight. 
How  much  a  pound  did  the  dressed  turkey  cost  me  ? 

22.  Divide  £  of  |  of  jf  of  |f  of  6J  by  f-  of  ^  of  fj  of  7f 

23.  Divide  |  of  tf  of  #  of  ft  of  8  J  by  f  of  ft  of  #  of  ff. 

24.  A  merchant  tailor  has  45f  yards  of  cloth,  from  which 
he  wishes  to  cut  an  equal  number  of  coats,  trousers,  and 
vests.     How  many  of  each  can  be  cut  from  it  if  the  gar- 
ments contain  4^  yards,  2j-  yards,  and  -f-  of  a  yard,  respec- 
tively ? 

25.  A  farmer  sold  at  market  20  sheep  at  $2-J  each,  and 
bought  8  yards  of  cloth  at  $!•§•  per  yard.     How  much 
money  had  he  left  ? 

26.  A  man  having  100  fowls,  sold  ^  of  them  to  E,  and 
-}  of  the  remainder  to  F.     What  was  the  value  of  what 
remained,  if  they  were  worth  25  cents  apiece  ? 

27.  How  many  tons  of  hay  will  be  required  to  keep  2 
horses  for  6  months,  if  8  horses  eat  15  tons  in  that  time  ? 


138  FRACTIONS. 

28.  A  walked  at  the  rate  of  3J  miles  an  hour  for 
hours,  and  B  at  the  rate  of  4|-  miles  an  hour  for  4^-  hours. 
Which  walked  the  greater  distance,  and  how  much  ? 

29.  A  young  man  received  $  1000  from  his  father.     He 
spent  -£$  of  it  for  clothes,  -|-  of  it  in  traveling,  and  invested 
the  rest  in  land.     How  much  did  he  pay  for  the  land  ? 

30.  Two  thirds  of  a  stock  of  goods  was  destroyed  by  fire, 
•|  of  the  remainder  was  destroyed  by  water,  and  the  rest  was 
sold  at  cost  for  $  2575.     What  was  the  cost  of  the  entire 
stock  ? 

31.  Five  eighths  of  A's  money  increased  by  the  difference 
between  -f  and  f  of  his  money  equals  $  1020.     How  much 
money  has  A  ? 

32.  A  pole  63  feet  long  was  broken  into  two  unequal 
pieces,  and  •§•  of  the  longer  piece  equaled  J  of  the  shorter. 
What  was  the  length  of  each  piece  ? 

SOLUTION —  Since  f  of  the  longer  piece  =  f  of  the  shorter  piece,  £  -of 
the  longer  piece  =  i  of  f ,  or  %  of  the  shorter  piece. 

Since  £  of  the  longer  piece  =  £  of  the  shorter,  the  longer  piece  =  f  of 
the  shorter  piece. 

Since  the  longer  piece  was  f  of  the  shorter,  both  pieces  must  have 
been  1|  or  £  times  the  shorter  piece,  which  is  63  feet. 

Since  f  of  the  shorter  piece  =  63  feet,  £  of  the  shorter  piece  was  \  of 
63  feet,  or  7  feet,  and  the  entire  length  of  the  shorter  piece  was  28  feet. 

63  feet  — 28  feet =35  feet,  the  length  of  the  longer  piece. 

33.  Divide  Jf.  of  Jf  of  ft  of  ?t  by  £  of  |  of  15. 

34.  I  have  a  farm  of  80  acres.     On  f  of  the  farm  corn  is 
planted,  on  -|  of  the  remainder  wheat,  and  on  the  rest,  oats. 
How  many  acres  are  there  of  each  kind  of  grain  ? 

35.  The  length  of  a  room  is  15f  feet,  and  the  width  is 
12J  feet.     What  will  be  the  cost  of  a  moulding  extending 
entirely  around  it  at  4J-  cents  a  foot  ? 


REVIEW   EXERCISES.  139 

36.  A  owned  f  of  a  ship  and  sold  f  of  his  share  to  B ;  B 
sold  |  of  what  he  bought  to  C  for  $2500.     At  that  rate 
what  was  the  whole  ship  worth  ? 

37.  From  4f  acres  I  sell  to  E  J.  of  an  acre,  to  F  f  of  an 
acre,  to  G  |-  of  an  acre,  and  to  H  -fa  of  an  acre.     How  much 
is  the  remainder  worth  at  $  850  per  acre  ? 

38.  A  gentleman  paid  $  60  for  keeping  2  horses  12  weeks. 
What  would  it  cost,  at  the  same  rate,  to  keep  one  horse  3% 
weeks  ? 

39.  Two  brothers  together  own  ^  of  a  flouring  mill  valued 
at  $  12520.     One  owns  -f-  as  much  as  the  other.     What  is  the 
value  of  each  one's  share  ? 

40.  A  lady  has  $  55f  in  her  purse.     She  spends  $  13^  for 
a  shawl,  $4f  for  cloth,  $3^  for  a  hat,  and  $2^  for  lace. 
How  much  has  she  left  ? 

41.  A  stove  manufacturer  purchases  old  iron  at  $-ff-$ 
per  hundred  pounds,  and  makes  out  of  it  stoves  weighing 
125  pounds  each,  which  he  sells  at  $  15^  apiece.     How  much 
does  he  gain  on  each  100  pounds  of  old  iron  ? 

42.  A  carpenter  alone  can  build  a  shop  in  18  days,  and 
with  the  help  of  his  son  he  can  build  it  in  12  days.     In  how 
many  days  can  the  son  alone  build  the  shop  ? 

43.  A  man  who  had  spent  f  of  his  money  and  $  ^  more, 
found  he  had  $  21  left.     How  much  money  had  he  at  first  ? 

44.  Simplify 

45.  I  bought  26  yards  of  carpet  at  $  l-^  a  yard,  3  curtains 
at  $  5f  each,  and  6  chairs  at  $  If  each.    What  was  my  bill  ? 

46.  A  huckster  bought  75  bushels  of  apples  at  $  |  per 
bushel,  65  bushels  at  $  f  per  bushel,  and  20  bushels  at  $  ±% 
per  bushel.     At  what  average  price  per  bushel  must  he  sell 
them  to  gain  $  25  ? 


140  FRACTIONS. 

47.  A  and  B  hire  a  pasture  for  $  15.50.     A  puts  in  8  cows, 
and  B  puts  in  12  cows.     What  must  each  pay  ? 

48.  Two  men  hire  a  pasture  for  $  20.     The  one  puts  in 
9  horses,  and  the  other  puts  in  48  sheep.     If  18  sheep  eat 
as  much  as  3  horses,  what  must  each  man  pay  ? 

49.  A  can  walk  a  mile  in  ^  of  an  hour,  and  B  in  ^-  of  an 
hour.     In  a  race  of  15  miles,  which  will  win,  and  by  how 
much? 

50.  James,  William,  and  Joseph  can  remove  a  pile  of 
wood  in  12  hours.     James  and  William  together  can  remove 
it  in   18   hours.      In  how  many  hours  can  Joseph  alone 
remove  it  ? 

51.  A  flour  dealer  bought  130  barrels  of  flour  at  $  6-f  per 
barrel.      He  sold  85  barrels  at  $6-^-  per  barrel,  and  the 
remainder  at  $  7.     What  did  he  gain  ? 

52.  A  and  B  together  have  $9500.     Two  thirds  of  A's 
money  equals  f  of  B's.     How  much  money  has  each  ? 

53.  A  lady  divided   $7J-  among  some  children,  giving 
them  $  y9^  each.     What  was  the  number  of  children  ? 

54.  E,  O,  and  W  bought  a  drove  of  cattle.     E  paid  for 
f  of  the  drove,  O  for  ^  of  it,  and  W  for  the  remainder.     It 
was  found  that  E  paid  $  56  more  than  W.     What  did  each 
pay,  and  what  was  the  cost  of  the  drove  ? 


56.  A  farmer  brought  to  market  3  jars  of  butter,  weigh- 
ing 27,  29,  and  40  pounds,  respectively.     The  empty  jars 
weighed  4J-,  4f,  and  1\  pounds.      The  butter  was  sold  for 
$  28.     What  was  the  price  per  pound  ? 

57.  A  man  bequeathed  to  his  son  $7500,  which  was  3{- 
times  what  he  gave  his  daughter.     What  did  the  daughter 
receive  ? 


REVIEW  EXERCISES.  141 

58.  A  owns  -f-  of  a  mill,  and  B  the  remainder;  f  of  the 
difference  between  their  shares  is  valued  at  $  8500.     What 
is  the  value  of  the  mill  ? 

59.  What  is  the  value  of  171  bales  of  cotton,  each  weigh- 
ing 4£  hundred-weight,  at  $  18f  per  hundred-weight  ? 

60.  A  spends  f  of  his  income,  and  B,  having  the  same 
income,  spends  1^  times  as  much  as  A,  and  finds  himself 
$  75  in  debt  at  the  end  of  the  year.     What  is  the  income 
of  each  ? 

61.  I  bought  75  acres  of  land  at  $63  an  acre.     I  sold  -^ 
of  it  at  $  71  an  acre,  -^  at  $  65  an  acre,  and  the  remainder 
for  $  3^-  more  per  acre  than  I  paid  for  it.     What  did  I  gain 
on  the  whole  ? 

62.  A  farmer  sold  320f  pounds  of  maple  sugar  at  15f 
cents  a  pound,  and  took  his  pay  in  cloth  at  67|  cents  a  yard. 
How  many  yards  did  he  receive  ? 

63.  A  ship  is  worth  $  85000.     A  man  owns  ^  of  it.     If 
he  sells  -f  of  his  share,  what  is  the  value  of  the  part  of  his 
share  which  is  left  ? 


64.  The  water  flows  from  one  spring  at  the  rate  of 
gallons  in  11  minutes  ;  from  another  spring  at  the  rate  of 
113  gallons  in  19  minutes.     Which  spring  flows  the  faster, 
and  what  is  the  difference  in  the  flow  per  minute  ? 

65.  A  merchant  sold  a  quantity  of  sugar  for  $1180,  and 
thereby  gained  \  of  the  cost.     If  he  had  sold  it  for  $  1000, 
would  he  have  gained  or  lost,  and  how  much  ? 

66.  What  is  the  exact  value  of  (3+2£—  f  of  f  +  f\-t-  4£  ? 

\  ¥/ 

67.  A  owes  $1140.     By  saving  -f$  of  his  income  annu- 
ally for  5  years,  he  can  pay  his  debt  and  have  $  1260  left. 
What  is  his  yearly  income  ? 


142  FRACTIONS. 

68.  A  man  has  three  creditors.      To  the  first  he  owes 
$1360£,  to  the  second   $.1087^,  and  to  the  third  $876f. 
The  man  fails,  and  the  creditors  seize  all  his  property,  which 
amounts  to  only  $  2350.     How  much  should  each  creditor 
receive  ? 

69.  Two  men  cleared  a  piece  of  woodland  for  $  69.     The 
one  worked  19  J  days  and  cut  71f  cords;  the  other  worked 
twice  as  many  days  as  the  first  cut  cords  per  day.     How 
much  did  each  receive,  if  they  shared  in  proportion  to  the 
time  they  worked  ? 

70.  A  stock-broker  bought  7  shares  of  Northern  Pacific 
Railroad  stock  at  $  97-J  per  share,  and  15  shares  of  Union 
Pacific  Eailroad  stock  at  $  lOlf  per  share.     He  sold  them 
all  at  $  105^  a  share.     How  much  did  he  gain  ? 

71.  If  a  miller  takes  -|  for  toll,  and  a  bushel  of  wheat 
produces  40  pounds  of  flour,  how  many  bushels  must  be 
carried  to  the  mill  to  obtain  196  pounds  of  flour,  or  1  barrel  ? 

72.  A  merchant  bought  three  pieces  of  cloth  for  $365f. 
The  first  contained  31-J-  yards,  the  second  42^-  yards,  and  the 
third  47f  yards.     He  wishes  to  sell  the  cloth  so  as  to  gain 
^  of  the  cost.     At  what  price  must  he  sell  it  per  yard  ? 

73.  An  estate  was  divided  between  two  brothers  and  a 
sister.      The   elder  brother  received  ^  of  the  estate,  the 
younger  ^,  and  the  sister  the  remainder,  which  was  $  1623 
more  than  the  younger  brother  received.      What  was  the 
value  of  the  estate,  and  what  did  each  receive  ? 

74.  Divide  9  times  the  product  of  ?i  and  ?  of  £  by  i$. 

2f  20       J  5£ 

75.  A  and  B  can  perform  a  piece  of  work  in  23-j-J  days. 
A  alone  can  perform  it  in  37J  days.     In  how  many  days 
can  B  do  the  work  alone  ? 


REVIEW  EXERCISES.  143 

76.  A  boat  whose  rate  of  sailing  in  still  water  is  14  miles 
an  hour,  was  accelerated  3J-  miles  per  hour  in  going  down- 
stream, and  retarded  the  same  distance  per  hour  in  coming 
Tip.     How  long  would  it  take  the  boat  to  come  up  the  same 
distance  that  it  could  go  down  in  10  hours  ? 

77.  Eight  men  cut  97  J  cords  of  wood  in  4J  days,  for 
which  they  received  $  48J.     What  was  the  average  daily 
pay  of  each  man  ? 

78.  Two  men  dug  a  ditch  for  $  75  ;  one  man  worked  3 J 
days  and  dug  14  rods ;  the  other  worked  as  many  days  as 
the  first  dug  rods  per  day.     How  much  did  each  receive,  if 
they  shared  in  proportion  to  the  time  they  worked  ? 

79.  A  man  has  -^  of  his  property  invested  in  real  estate, 
$  of  the  remainder  in  bonds,  £  of  what  still  remains  in  bank 
stock,  and  the  rest,  which  is  $3500,  he  has  invested  in 
business.     What  is  the  value  of  his  entire  property  ? 

80.  The  sum  of  two  numbers  equals  5— 2,  and  one  of 

them  is  the  difference  between  — 1£  and  — 5.    What  is  the 
other  number  ? 

81.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
If  A  can  do  only  J  as  much  as  B,  how  long  will  it  take  each 
of  them  to  do  the  work  ? 

82.  A  man  bought  a  house  and  paid  ^  of  the  price  in 
cash  at  the  time  of  the  purchase.    A  year  afterward  he  paid 
-J  of  what  then  remained  unpaid,  and  the  two  payments 
amounted  to  $  5260.     How  much  did  he  pay  for  the  house  ? 

83.  A  cistern  which  holds  280  gallons  is  empty.     It  has 
a  supply  pipe  which  will  fill  it  in  10  hours,  and  a  discharge 
pipe  which  will  empty  it  in  7  hours.     If  the  supply  pipe 
has  been  running  into  it  for  4  hours,  and  then  both  pipes 
are  opened,  'in  what  time  will  it  be  emptied  ? 


DECIMAL   FRACTIONS. 


164.  1.    What  is  1  of  the  ten  equal  parts  of  anything  ? 

2.  What  is  1  of  the  ten  equal  parts  of  -^  ?     3  parts  ? 

3.  What  is  1  of  the  ten  equal  parts  of  -^  ?     35  parts  ? 

4.  What  is  1  of  the  ten  equal  parts  of  -3-^-5-  ?     43  parts  ? 

5.  What  part  of  TV  is  y^?    Of^is^?    Of  ^  is 
i  o  o  o  o  • 

6.  What  part  of  ^  is  Tfo  ?     Of  ^  is  ^  ?     Of  yrfinr 
is  10000  ? 

165.  The  divisions  of  anything   into  tenths,  hundredths, 
thousandths,  etc.,  are  called  Decimal  Divisions. 

166.  One  or  more  of  the  decimal  divisions  of  a  unit  is 
called  a  Decimal  Fraction. 

The  word  decimal  is  derived  from  the  Latin  word  decem,  ten. 
Decimal  fractions  are  commonly  called  decimals. 

167.  Since  tenths  are  equal  to  ten  times  as  many  hun- 
dredths,  hundredths  equal  to  ten  times  as  many  thousandths, 
etc.,  decimals  have  the  same  law  of  increase  and  decrease 
as  integers,  and  the  denominator  may  be  indicated  by  the 
position  of  the  figures. 

In  the  decimal  system  of  notation,  the  representative 
value  of  a  figure  is  decreased  tenfold  by  each  removal  one 
place  to  the  right ;  hence  : 

The  figure  at  the  right  of  units  expresses  tenths. 
144 


NUMERATION.  145 

The  figure  at  the  right  of  tenths  expresses  hundredths. 

The  figure  at  the  right  of  hundredths  expresses  thousandths. 

The  figure  at  the  right  of  thousandths  expresses  ten- 
thousandths. 

The  figure  at  the  right  of  ten-thousandths  expresses  hun- 
dred-thousandths, etc. 

168.  A  period,  called  the  Decimal  Point,  is  placed  before 
the  decimal. 

Thus,  .5  represents  T5<j-  ;  .68  represents  r5^. 

.0007  = 


.05  =  ^        .004  =  Tlftnr        .1461  = 
NUMERATION  TABLE. 


1 
\ 

g 

CO 

X 

USANDTHS. 

o 

Q 

Z 

co 

CO 

z 

O 

I 

DRED-Th 

-THOUSA 

USANDS. 

% 

Ul 

a: 
Q        co 

CO 

•       x 
£        h 

DREDTH 

1 

-THOUSA 

Q 
Ul 

cc. 

Q 

.IONTHS. 

J 

Z 

Z 

O 

z       z 

E       z 

Z 

o 

Z 

Z 

3 

Ul 

I 

Z          Ul 

Ul 

D 

i 

X 

h 

h 

X        H 

D        h 

X 

1- 

h 

X 

i 

3 

4 

1 

5 

6    7 

3.4 

1 

3 

2 

0 

7 

The  number  is  read,  3  million,  416  thousand,  673  and  413  thousand 
207  millionths. 

The  orders  of  decimals  below  millionths  are  ten-millionths,  hundred- 
millionths,  billionths,  ten-billionths,  hundred-billionths,  trillionths,  etc. 

EXERCISES  IN  NUMERATION. 
169.   1.   Kead  the  expression  5.239. 
The  whole  expression  is  read,  5  and  239  thousandths. 

STAND.    AR.  —  10 


146  DECIMAL  FRACTIONS. 

RULE.  —  Read  the  decimal  as  an  integral  number,  and  give 
it  the  denomination  of  the  right-hand  figure. 
Read  the  following : 

2.  .324.                 9.     323.56.              16.  385.043685. 

3.  .457.               10.     4.5283.              17.  42.0004367. 

4.  .3856.              11.   46738.4.              18.  916.380043. 

5.  .2834.              12.   50.0037.              19.  74348.0435. 

6.  .5169.              13.   310.009.              20.  8.04045006. 

7.  .0045.              14.   346.009.              21.  91873.0009. 

8.  .0008.              15.    92080.9.             22.  90.90900009. 

EXERCISES  IN   NOTATION. 

170.   1.   Express  decimally  seventy-nine  thousandths. 

EXPLANATION.  —  Since  thousandths  occupy  the  third  place,  three 
figures  are  required  to  express  the  decimal.  Hence,  the  number 
seventy-nine  is  written,  and  a  cipher  prefixed  to  cause  the  figures  to 
occupy  their  proper  position.  Hence,  the  decimal  is  written  .079. 

RULE.  —  Write  the  numerator  of  the  decimal,  prefix  ciphers 
if  necessary  to  indicate  the  denominator,  and  place  the  deci- 
mal point  before  tenths. 

Express  decimally : 

2.  Eight  tenths.      Three  tenths.      Five  tenths.      Pour 
hundredths.     Eight  hundredths.     Six  hundredths. 

3.  Five  thousandths.     Three  hundred  four  thousandths. 
Seven    ten-thousandths.       Eight    hundred   sixty-five   ten- 
thousandths.     Sixty-eight  thousandths. 

4.  Fifteen  hundredths.      Twelve   hundred-thousandths. 
Forty-eight  thousandths.     Four  hundred-millionths. 

5.  Ninety-five  ten-thousandths.  Ninety  billionths.  Sixty- 
two  hundred-thousandths.     Fifty-five  thousandths. 

6.  210  millionths.      403  thousandths.      15  hundredths. 
4256  hundred-thousandths.     1268  hundred-millionths. 


NOTATION.  14T 

7.  56345  millionths.     389  ten-thousandths.     3854  hun- 
dred-thousandths.    518  ten-millionths. 

8.  Sixty-seven,  and  three  hundred  forty-nine  ten-thou- 
sandths.    Fifty,  and  fifty-five  millionths. 

9.  Eighty-eight,  and  five  thousand   five  hundred-thou- 
sandths.    Nine,  and  nine  ten-thousandths. 

10.  TV  13.    TV5^.         16.    A.  19.        . 

11.  ^  14.    ^frjhnr.       17.    3%.  20. 

12.  ^.  15.    5T\  18.    ^%.  21. 

22.  How  do  the  fractions  in  16,  17,  and  18  compare  in 
value  ?      How  do  the  decimals   compare  ?      What  is  the 
effect  of  annexing  ciphers  to  decimals  ? 

23.  How  do  the  fractions  in  19,  20,  and  21  compare  in 
value?      How   do  the  decimals  compare?      What  is  the 
effect  of  prefixing  decimal  ciphers  to  a  decimal  ? 

24.  How  does  the  number  of  places  in  a  decimal  com- 
pare with  the  number  of  ciphers  in  the  denominator. 

171.  PRINCIPLES.  — 1.   Annexing  ciphers  to  a  decimal  does 
not  alter  its  value. 

2.  Each  decimal  cipher,  prefixed  to  a  decimal,  divides  the 
value  of  the  decimal  by  ten. 

3.  The  denominator  of  a  decimal,  when  expressed,  is  1  with 
as  many  ciphers  annexed  as  there  are  figures  in  the  decimal. 

172.  In  expressions  of  the  currency  of  the  United  States, 
the  cents,  mills,  etc.,  may  be  read  as  decimals  of  a  dollar  : 

Thus,  $5.375  may  be  read  5  dollars  and  375  thousandths,  or  5  dol- 
lars 37T5^  cents. 

Eead  the  following  as  dollars  and  decimals  of  a  dollar : 

1.  $6.495.  3.  $7.394.  5.  $4.004. 

2.  $5.083,  4.  $5.865.  6.  $8.056. 


148  DECIMAL  FRACTIONS. 

REDUCTION. 

173.  To  reduce  dissimilar  to  similar  decimals. 

1.  Eeduce  .5,  .25,  .046,  and  2.0506  to  similar  decimals. 

.5         =    .5000  EXPLANATION.  —  Since  the  lowest  order  of  deci- 

25      =     2500  mals  in  tlie  &iven  numbers  is  ten-thousandths, 

'                   OAKO  a11  tlie  Decimals  must  be  changed  to  ten-thou- 

.U4b    —    .U4bU  san(iths.     This  may  be  done  by  annexing  ciphers 

2.0506  =  2.0506  (Prin.  1,  Art.  171). 

KULE.  —  Give  all  the  decimals  the  same  number  of  places  by 
annexing  ciphers. 

Reduce  to  similar  decimals : 

2.  .4,  .65,  .175.  8.  .3460,  .17,  .4,  17.6. 

3.  .05,  .015,  .75,  .0104.  9.  4.08,  .7,  .0004. 

4.  .045,  .476,  .00055.  10.  9,  .9,  .009,  90. 

5.  .0043,  .1,  .140,  .07865.  11.  6.1,  1.054,  12.36876. 

6.  1.2,  .43,  .105,  .10017.  12.  75,  4.1,  .268,  .0057. 

7.  3.27,  .0005,  .584.  13.  100,  .001, 1000,  .000001. 

174.  To  reduce  decimals  to  common  fractions. 

1.  Eeduce  .75  to  a  common  fraction. 

EXPLANATION.  —  .75  expressed  as  a  common  frac- 
-^  =  1%" =  f     ti°n  i8  T5^  or  1 5  when  it  is  reduced  to  its  lowest 
terms. 

RULE.  —  Omit  the  decimal  point,  supply  the  proper  denomi- 
nator, and  reduce  the  fraction  to  its  lowest  terms. 

2.  .25.  4.   .65.  6.   .38.  8.   .375. 

3.  .75.  5.   .52.  7.   .64.  9.   .875. 


REDUCTION.  149 

10.  .435.  14.  .0375.  18.  .7024.  22.  .05165. 

11.  .568.  15.  .0215.  19.  .0475.  23.  .01235. 

12.  .405.  16.  .1135.  20.  .05625.  24.  .41275. 

13.  .635.  17.  .0025.  21.  .03435.  25.  .00015. 

26.    Reduce  .87^-  to  a  common  fraction. 

EXPLANATION.  —  The  expression 

071  _  87-|-  __  —  14.5  _  1.      written  as  a  common  fraction  be- 

•°  '  "2"  —  ^  f\(\      1  r\f\     Ttnr      s  *  14.5. 

comes  -^-.     Performing  the  divi- 

sion indicated,  or  reducing  the  denominator  also  to  halves,  it  becomes 
Hft.  °r  f  . 

Reduce  to  common  fractions  : 


27.    .12f 

31.    .62£. 

35.    .416|. 

39.    .00141-. 

28.    .18f. 

32.    .41^. 

36.    .003f. 

40.   12.095f 

29.    .371 

33.    .24f. 

37.    .075TV 

41.   22.71|. 

30.    .49|. 

34.    .56f. 

38.    .643f 

42.   43.87|. 

175.   To  reduce  a  common  fraction  to  a  decimal. 

1.  How  many  tenths  are  there  in  1  ?     In  £  ?    In  f  ?    In 
|?     In|? 

2.  How  many  hundredths  are  there  in  1  ?     In  J  ?  In  f  ? 
Ini?     In  |? 

3.  How  many  hundredths  are  there  in  3  ?     In  £  of  3, 
orf  ? 

4.  How  many  hundredths  are  there  in  4?     In  -J-  of  4, 
orf? 

5.  How  many  thousandths  are  there  in  1  ?     In  £  ? 

6.  How  many  thousandths  are  there  in  2  ?     In  -|  ? 


150 


DECIMAL  FRACTIONS. 


7.   Reduce  j-  to  an  equivalent  decimal. 

8")  7  000          EXPLANATION.  —  f  is  |  of  7.  In  7  there  are  70  tenths,  and 

}  I  of  70  tenths  is  8  tenths  and  6  tenths  remainder.    6  tenths 

.o7o     are  equai  to  60  hundredths,  and  |  of  60  hundredths  is  7 

hundredths  and  4  hundredths  remainder.     4  hundredths  are  equal  to 

40  thousandths,  and  \  of  40  thousandths  is  5  thousandths.     Hence  $  is 

equal  to  8  tenths  +  7  hundredths  +  5  thousandths,  or  .875. 

RULE.  —  Annex  ciphers  to  the  numerator  and  divide  by  the 
denominator.  Point  off  as  many  decimal  places  in  the  quotient 
as  there  are  ciphers  annexed. 

1.  In  many  cases  the  division  is  not  exact.     In  such  instances  the 
remainder  may  be  expressed  as  a  common  fraction,  or  the  sign  +  may 
be  employed  after  the  decimal  to  show  that  the  result  is  not  complete  ; 
thus  £  = .  166f ,  or  .  166  -f. 

2.  Common  fractions  in  their  lowest  terms  cannot  be  reduced  to 
exact  decimal  values  or  pure  decimals  when  their  denominators  con- 
tain any  prime  factors  besides  2  or  5.     This  truth  is  evident,  from  the 
fact  that  when  ciphers  are  annexed  to  the  numerator,  that  is,  when  it 
is  multiplied  by  10,  only  the  factors  2  and  5  are  introduced  into  the 
numerator;   consequently  if  any  other  prime  factor  is  found  in  the 
denominator,  the  division  cannot  be  exact. 

3.  It  is  evident  also  that  fractions  whose  denominators  are  com- 
posed of  the  factors  2  or  5  have  exact  decimal  values. 

Change  the  following  to  decimals  : 


8. 

f 

17. 

f 

26.   ^. 

35. 

» 

44. 

37f 

9. 

i- 

18. 

<&• 

27.    |f 

36. 

*f 

45- 

.451 

10. 

f 

19. 

f 

28.    ft. 

37. 

H- 

46. 

16.4f 

11. 

f. 

20. 

» 

29.    ^. 

38. 

ff 

47. 

48.5^. 

12. 

i- 

21. 

A- 

30.    Th- 

39. 

12f 

48. 

.23f 

13. 

A* 

22. 

A- 

31-    it- 

40. 

18f. 

49. 

60.0f 

14. 

it- 

23. 

tt- 

32.    |f 

41. 

24f 

50. 

.OOQ|f 

15. 

f 

24. 

yV 

33.    ~g~(j"» 

42. 

if- 

51. 

513.00|. 

16. 

t 

25. 

•fr 

34.    ^. 

43. 

ih- 

52. 

75.0001 

ADDITION.  151 

ADDITION. 
176.   1.   What  is  the  sum  of  .29,  3.314,  41.2356  ? 

.29  EXPLANATION.  —  The  numbers  are  written  so  that  units 

o  01  A        of  the  same  order  stand  in  the  same  column,  and  they  are 
added  precisely  as  in  integers.     The  decimal  part  of  the 
O     gum  ig  separated  from  the  integral  part  by  the  decimal 


44.8396    P°mt- 

The  decimals  may  be  made  similar  by  annexing  ciphers 

until  all  the  decimals  have  the  same  number  of  places,  and  then 
added  ;  but  this  is  not  commonly  done. 

Find  the  sum  of  the  following  : 

2.  3.25,  1.6,  32.043,  .341,  15.        5.  .004,  5.75,  .026,  4.1. 

3.  1.14,  70,  .014,  .6413,  24.          6.  6.845,  .137,  2.5,  .1004. 

4.  .45,  .076,  41.7,  .0457.  7.  .964,  .0034,  46,  7.37,  .08. 

8.  .125,  1.25,  12.5,  125,  .0125. 

9.  37,  5.4,  62.5,  .44,  3.845. 

10.  4.2,  .034,  78.9,  62.5,  148.9. 

11.  12.34,  1.2,  16.5,  27.4,  15.35,  174.8. 

12.  4.1,  67.5,  42.001,  13.18,  .0004. 

13.  146.9,  .00412,  31.416,  125.001,  231.8. 

14.  47.25,  5.00695,  193.5,  5.875,  9.0000105. 

15.  $7.28,  $213.09,  $.21,  $13.42,  $.15. 

16.  $10.25,  $8.95,  $3.02,  $135.24,  $185.64. 

17.  $200,  $.20,  $2.05,  $.12|,  $3.18f. 

18.  $1.35,  $16.50,  $2.37£,  $.56J,  $2000. 

19.  A  family  used  .85  of  a  ton  of  coal  in  January,  .75  of 
a  ton  in  February,  .675  of  a  ton  in  March,  and  .5  of  a  ton  in 
April.  How  many  tons  of  coal  did  they  use  ? 


152  DECIMAL  FRACTIONS. 

20.  A  man  earned  $  6.75  in  one  week,  $  7.25  in  another, 
$7.37^-  in  another,  $8.12^-  in  another,  and  $9  in  another. 
How  much  did  he  earn  in  the  five  weeks  ? 

21.  In  four  piles  of  wood  there  are  respectively  5.316 
cords,  8J  cords,  12.25  cords,  and  13.569  cords.     How  many 
cords  are  there  in  all  the  piles  ? 

22.  Find  the  sum  of  three  hundred  and  five  hundredths, 
two  thousand  one  hundred  eight  and  four  thousandths,  three 
millionths,  one  hundred  seventeen  thousand  seven  hundred 
seven  and  forty-five  millionths. 

23.  A  bookseller  bought  5  complete  sets  of  copy-books 
for  $  3.60,  50  geographies  for  $  50,  25  physiologies  for  $  25, 
30  grammars  for  $14,  and  55  spellers  for  $9.90.     How 
much  did  all  the  books  cost  him  ? 

24.  Find  the  sum  of  two  thousand  three  hundred  one 
and   thirty-nine   hundredths,   three  tenths,   two    thousand 
seven  hundred   forty-nine   ten-thousandths,   and    thirteen 
thousandths. 

25.  A  lady  bought  5  dozen  buttons  for  $1.08,  2  yards  of 
ribbon  for  $.37-^-,  16  yards  of  muslin  for   $1.18f,   some 
thread  and  needles   for  $.31^,   and  a  dress  for   $8.62£. 
What  was  the  amount  of  her  purchases  ? 

26.  Find  the  sum  of  two  and  five  ten-thousandths,  forty- 
three  thousandths,  sixty-three  and  four  hundred  fifteen  hun- 
dred-thousandths, and  five  hundred  thirteen  ten-thousandths. 

27.  A  man  bought  a  house  for  $4000,  a  store  for  $3780, 
merchandise  for  $12751.85,  a  horse  for  $185.80,  a  farm  for 
$6175,  and  bank  stock  $5760.56.      What  did  the  whole 
cost  him  ? 

28.  Mr.  Clarke  bought  for  his  house  one  set  of  parlor 
furniture  for  $1234.69,  carpets  for  $345.97,  a  piano  for 
$500,  a  mirror  for  $29.75,  curtains  for  $132.19,  and  oil 
paintings  for  $  8975.43.     How  much  did  he  pay  for  all  ? 


SUBTRACTION. 


153 


SUBTRACTION. 
177.   1.   From  57.25  subtract  33.1468. 


57.2500 
33.1468 


EXPLANATION.  —  The  numbers  are  written  so  that  units 
of  the  same  order  stand  in  the  same  column,  and  they  are 
subtracted  as  in  integers.     The  decimal  part  of  the  re- 
24.1032     mainder  is  separated  from  the  integral  part  by  the  decimal 

point. 

The  decimals  may  be  made  similar  by  annexing  ciphers  until  both 
have  the  same  number  of  places,  but  the  ciphers  may  be  supposed  to 
be  there  even  though  they  are  not  written. 


Find  the  value  of : 

2.  .325 -.106. 

3.  .4806 -.3124. 

4.  3.5872-1.2834. 

5.  5.4618-3.2403. 

6.  17.8465-5.6341. 

7.  315.42346-10.326. 

8.  34.832-18.068193. 

9.  125.4276-19.305. 

10.  48.76-30.428. 

11.  72.154-61.075. 

12.  476-245.75. 

13.  355.8-196.954. 

14.  750-84.1206. 


15.  647.625  -  .995. 

16.  1000 -.001. 

17.  $34.185 -$8.27. 

18.  $  45.67 -$  18.50. 

19.  $  63.10 -$  27.43. 

20.  $75.35 -$49.75. 

21.  $88.125 -$1.875. 

22.  $125.75 -$67.50. 

23.  $11.10 -$1.14. 

24.  $100-$.37£. 

25.  $189.46f-$7.62i 

26.  $225.87£-$175.65f. 

27.  $  437.83| -$216.241 


28.  A  vessel  sailed  from  Portland,  Me.,  for  New  Orleans, 
with,  a  cargo  of  1528.375  tons  of  ice.  On  the  way  94.85  tons 
of  it  melted.  How  much  ice  reached  New  Orleans  ? 


154  DECIMAL  FRACTIONS. 

29.  A  man  bought  a  suit  of  clothes  for  $  35.50,  a  hat  for 
$  3.75,  and  a  pair  of  gloves  for  $  1.87.     He  gave  the  sales- 
man a  hundred-dollar  bill.     How  much  change  ought  he  to 
have  received  ? 

30.  From  five  hundred  eighty  and  sixty-seven  ten  thou- 
sandths take  ninety-six  and  forty-nine  millionths. 

31.  A  merchant  bought  a  tub  of  butter  for  $  14.62^,  pay- 
ing $  7.87-J  in  cloth,  $4.58  in  groceries,  and  the  rest  in 
money.     How  much  money  did  he  pay  ? 

32.  A  butcher  killed  an  ox  that  cost  him  $  56.75.     He 
retailed  the  meat  for  $  54.28,  sold  the  tallow  for  $  4.95,  and 
the  hide  for  $  7.65.     What  were  his  profits  ? 

33.  A  merchant  had,  at  the  beginning  of  the  year,  goods 
worth  $7600.      During  the  year  he  bought  goods  to  the 
amount  of  $  6735.75,  and  sold  to  the  amount  of  $  9875.84. 
At  the  close  of  the  year  his  inventory  showed  goods  on  hand 
worth  $  7026.65.    How  much  did  he  make  during  the  year  ? 

34.  In  a  cistern  that  will  hold  326.5  barrels  of  water, 
there  are  178.625  barrels.     How  much  does  it  lack  of  being 
full? 

35.  A  man  owned  sixty-nine  hundredths  of  a  township 
of  land,  and  sold  sixty-nine  thousandths  of  the  township. 
How  much  did  he  still  own  ? 

36.  If  I   spend  $45.89J   for   merchandise,   how   much 
change  will  I  receive  from  a  fifty-dollar  bill  ? 

37.  The  receipts  of  a  factory  for  a  certain  year  were 
$  1,374,837.64  and  the  expenses  were  $  1,100,095.75.    What 
were  the  profits  ? 

38.  A  man  whose  income  was  $15,745  spent  one  year 
$  12,349.97.     How  much  did  he  save  that  year  ? 


MULTIPLICATION.  155 

MULTIPLICATION. 

178.   1.   -nr  x  ro^Tinr  5  TCIT  x  TTF=  i  o  o  o  5  iooox  TTT^  i  o  o  o  oa 


2.  How  does  the  number  of  ciphers  in  the  denominator 
of  the  product  compare  with  the  number  of  ciphers  in  the 
denominators  of  the  factors  ? 

3.  How  does  the  number  of  places  in  a  decimal  compare 
with  the  number  of  ciphers  in  its  denominator  ? 

4.  How  many  places,  then,  will  there  be  in  the  product 
of  two  decimals  ? 

179.  PRINCIPLE.  —  The  product  of  two  decimals  contains 
as  many  decimal  places  as  there  are  decimal  places  in  both 
factors. 

WRITTEN   EXERCISES. 

180.  1.   What  is  the  product  of  .417  multiplied  by  .34  ? 

.417          EXPLANATION.  —  The   numbers   may    be    multiplied   as 
.34     though  they  were  integers.     Since  the  multiplier  contains  2 
decimal  places,  and  the  multiplicand  3  decimal  places,  the 


1251  Pr°duct  will  contain  5  decimal  places,  and  the  decimal  point 
-  is  placed  before  the  fifth  figure  counting  from  the  right 
.14178  (Prm.). 

RULE.  —  Multiply  as  if  the  numbers  were  integers,  and  from 
the  right  of  the  product  point  off  as  many  figures  for  decimals 
as  there  are  decimal  places  in  both  factors. 

If  the  product  does  not  contain  as  many  figures  as  there  are  deci- 
mals in  both  factors,  the  deficiency  must  be  supplied  by  prefixing 
ciphers. 

Find  the  product  of: 

2.  .25  x  .32.  4.   .126  x  35. 

3.  .48  x  4.8.  5.   .043  x  6.5. 


156  DECIMAL  FRACTIONS. 

6.  348  x  .46.  25.      63.18  x  2.402. 

7.  .0432  x  5.4.  26.      51.27  x  5.321. 

8.  34.8  x. 74.  27.   24.075  x  16^. 

9.  .048  x  24.  28.        450  x  .06. 

10.  50  x. 008.  29.        .045  x  18£. 

11.  .095x40.  30.         64.  x. 032. 

12.  3.24x3.3.  31.       30.3  x  .024. 

13.  255  x. 0007.  32.        .046x25. 

14.  6.75  x8f.  33.     3.826  x6f 

15.  34.5x11.2.  34.  37.555x45.64. 

16.  8.75x8.5.  35.  3.005x25.4. 

17.  .759  x  .032.  36.  214.76  x  89.104. 

18.  436  x  2.75.  37.  .04128  x  .00025. 

19.  3.45x6.24.  38.  4.2008x1.25. 

20.  347  x  .085.  39.      34.10  x  12.6. 

21.  5.6  x. 056.  40.    185.75  x!64f. 

22.  3.75  x  121*  41-    -04261  X  31245. 

23.  35.16  x5|.  42.      87.03  x  8.412. 

24.  50.05  x. 045.  43.    14.136  x  .00045. 

44.   Multiply  7.5864  by  200. 

EXPLANATION.  —  Since  each  removal  of  a  figure  one 
7.5864         place  to  the  left  increases  its  value  tenfold,  the  removal 
200     of  the  decimal  point  one  place  to  the  right  multiplies  by 
10,  and  two  places  by  100.     The  product  of  7.5864  x  100 


1517.28          is  therefore  758.64,  and  this  multiplied  by  2  gives  the 
product  of  7.5864  x  200,  which  is  1517.28. 

45.  5.836  x  100.       49.  .3856  x  200.     53.  42.8364  x  3000. 

46.  16.834  x  100.       50.  .4937  X  300.     54.  876.423  x  4000. 

47.  95.817  x  1000.     51.  5.927  x  500.     55.  915.976  x  5000. 

48.  373.186  x  1000.     52.  59.47  x  600.     56.  .813426  x  6000. 


MULTIPLICATION.  157 

57.  What  will  be  the  cost  of  8.5  reams  of  paper  at  $  3.62  J 
a  ream  ? 

58.  How  much  must  be  paid  for  35.75  bushels  of  corn  at 
$  .625  per  bushel  ? 

59.  When  land  is  worth  $126.75  per  acre,  how  much 
must  be  paid  for  a  farm  of  65  acres  ? 

60.  How  many  yards  are  there  in  25  pieces  of  tapestry 
carpeting,  if  each  piece  contains  32.75  yards  ? 

61.  If  a  rolling-mill  makes  95.6  tons  of  iron  per  day,  how 
many  tons  will  it  make  in  142.25  days  ? 

62.  A  man  bought  3.5  yards  of  broadcloth  at  $  3.75  per 
yard,  4  yards  of  cashmere  at  $  .87-|  per  yard,  26  yards  of 
calico  at  $  .061  per  yard,  and  14  yards  of  muslin  at  $  .07 
per  yard.     What  was  the  cost  of  the  whole  ? 

63.  A  grocer  sold  28.5  pounds  of  sugar  at  5|-  cents  a 
pound,  and  22.6  pounds  of  lard  at  7^-  cents  a  pound.     How 
much  did  he  receive  for  both  ? 

64.  Mr.  Ball  sold  65  bushels  of  wheat  at  $  1.12 J  a  bushel, 
27.4  bushels  of  clover  seed  at  $4.37-1-  a  bushel,  and  180 
bushels  of  corn  at  $  .62^  a  bushel.     How  much  did  he  get 
for  the  whole  ? 

65.  A  mechanic  earned  $14.87£  a  week  for  4  weeks. 
The  first  week  he  spent  $  7.28,  the  second  week  he  spent 
$  9^-,  the  third  week  he  spent  $  6.25,  and  the  fourth  week 
he  spent  $  8-J-.     How  much  money  did  he  save  ? 

66.  A  farmer  sold  40  bushels  of  oats  at  $  .37^  a  bushel, 
and  35^  bushels  of  potatoes  at  $  .56  a  bushel.     He  received 
in  payment  25  pounds  of  sugar  at  $  .05^  a  pound,  4  pounds 
of  rice  at  $  .10  a  pound,  3  gallons  of  molasses  at  $  .65  a 
gallon,  and  the  balance  in  cash.    How  much  cash  did  he 
receive  ? 


158  DECIMAL  FRACTIONS. 


DIVISION. 

181.  1.    .6x.8  =  .48;  .6  x. 08  =  .048;  .6  x  .008  =  .0048. 

2.  How  many  decimal  places  are  there  in  the  product  of 
two  decimals  ? 

3.  If  the  product  and  one  of  the  factors  are  given,  how 
may  the  number  of  decimal  places  in  the  other  factor  be 
found  ? 

4.  Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  if  the  divisor  and  dividend  are  given,  how  may 
the  number  of  decimal  places  in  the  quotient  be  found  ? 

182.  PRINCIPLE.  —  The  quotient  will  contain  as  many  deci- 
mal places  as  the  number  of  decimal  places  in  the  dividend 
exceeds  those  in  the  divisor. 

WRITTEN   EXERCISES. 

183.  1.   Divide  .00864  by  .24. 

9A.\  nr»S£JY  OQA         EXPLANATION.  —  The  numbers  are  divided  as  if 
.Z4).UU     >4(.Ut5b     they  were  integers.     Since  the  dividend  contains 
72  5  decimal  places,  and  the  divisor  2,  the  quotient 

TTT  contains  5  —  2,  or  3  decimal  places  (Prin. ).    Since 

there  are  only  two  figures  in  the  quotient,  a  cipher 
144  is  prefixed  to  make  the  required  number  of  deci- 

mal places. 

EULE. — Divide  as  if  the  numbers  were  integers,  and  from 
the  right  of  the  quotient  point  off  as  many  figures  for  decimals 
as  the  number  of  decimal  places  in  the  dividend  exceeds  the 
number  of  those  in  the  divisor. 

1.  If  the  quotient  does  not  contain  a  sufficient  number  of  decimal 
places,  the  deficiency  must  be  supplied  by  prefixing  ciphers. 

2.  Before  commencing  the  division,  the  number  of  decimal  places 
hi  the  dividend  should  be  made  at  least  equal  to  the  number  of  decimal 
places  in  the  divisor. 


DIVISION. 


159 


3.  When  there  is  a  remainder  after  using  all  the  figures  of  the 
dividend,  annex  decimal  ciphers  and  continue  the  division. 

4.  For  the  ordinary  purposes  of  business,  it  is  not  necessary  to 
carry  the  division  further  than  to  obtain  four  or  five  decimal  figures  in 
the  quotient. 


Find  the  quotients  of : 

2.  34.75-5-25. 

3.  46.103-5-2.14. 

4.  2.450-5-9.8. 

5.  7.8125-5-31.25. 

6.  272.636-5-6.37. 

7.  .00335  --6.7. 

8.  6.2512 -5- .37. 

9.  $2756.25-5- $31.5. 

10.  .05475-5-15. 

11.  18.312-5-24. 

12.  16.025  -5-  .045. 

13.  105.70 -v- 3.5. 

14.  .11928 -5- .056. 

15.  112.1184-5-9.16. 

16.  9322.15-5-6.275. 

17.  .04905 -5- .327. 


19.  281.8585-5-3.85. 

20.  687.50 -5- .025. 

21.  $  68.875 -5- $  145. 

22.  34.368  -5-  .013. 

23.  .014532 -*- .0692. 

24.  3.72812  -5-  4.07. 

25.  18712.264-5-1.52. 

26.  .33615-5-12.45. 

27.  62.41  -*-. 079. 

28.  3.1812-5-482. 

29.  17.28-5-1728. 

30.  1728-5-17.28. 

31.  .00255-5-51. 

32.  75 -5- .0125. 

33.  $135-5-$  .371. 

34.  725.406 -5- .0957. 

35.  .0021318-5-38. 


18.   135.05  -5-  .037. 

36.   Divide  568.148  by  200. 

200"i  568  148         EXPLANATION.  —  Since  each  removal  of  a  figure 
'— - — '— —     one  place  to  the  right  decreases  its  value  tenfold, 
2.84074     the  removal  of  the  decimal  point  one  place  to  the 
left  divides  by  10,  and  two  places  by  100.     The  quo- 
tient of  568.148  -4- 100  is,  therefore,  5.68148,  and  this  divided  by  2  gives 
the  quotient  of  568.148  -r-  200,  which  is  2.84074. 


160  DECIMAL  FRACTIONS. 

Find  the  quotients  of : 

37.  165 -h  50.  44.  3725.4-^700. 

38.  48.250 -f- 20.  45.  569000 -r- 800. 

39.  382.476 -- 200.  46.  72.3450  -- 1000. 

40.  725.61-5-300.  47.  4624.12 -j- 2000. 

41.  59.60430 -:- 600.  48.  .51648  -h  3000. 

42.  7.645-5-500.  49.  128.7642  -,- 1200. 

43.  .94876 -h  400.  50.  3094.32 -- 15000. 

51.  If  a  man  earns  $162  in  13.5  weeks,  what  are  his 
average  wages  per  week  ? 

52.  At  $8.25  per  ton,  how  much  hay  can  be  bought  for 
$45.85? 

53.  At  $  10.50  each,  how  many  harrows  can  be  bought 
for  $178.50? 

54.  If  a  barrel  of  flour  costs  $5.75,  how  many  barrels 
can  be  bought  for  $258.75  ? 

55.  At  $.24  per  dozen,  how  many  dozen  eggs  can  be 
bought  for  $30.72? 

56.  If  $  640.05  are  paid  for  75.3  tons  of  coal,  what  is  the 
average  price  per  ton  ? 

57.  There  are  31.5  gallons  in  a  barrel.     How  many  bar- 
rels are  there  in  2787.75  gallons  ? 

58.  A  farmer  sold  22.5  bushels  of  wheat  at  $  .98  a  bushel, 
and  a  certain  number  of  bushels  of  corn  at  $  .625  a  bushel. 
He  received  for  his  corn  $  9.20  more  than  he  did  for  his 
wheat.     How  many  bushels  of  corn  did  he  sell  ? 

59.  At  $  18.75  each,  how  many  dressing  bureaus  can  be 
bought  for  $  506.25  ? 

60.  At  $  5.75  per  ton,  how  many  tons  of  range  coal  can. 
be  bought  for  $51.75? 


WRITTEN  EXERCISES. 


161 


SHORT   PROCESSES. 
WRITTEN   EXERCISES. 

184.   To  multiply  by  a  number  a  little  less  than  100,  1000, 
etc. 

1.  Multiply  6834  by  98. 

EXPLANATION.  —  98  times  a  number  is 

100  X  6834  =  683,400     100  times  the  numker  minus  2  times  the 

2  X  6834  =    13,668     number.     Hence  the  number  may  be  mul- 

98  X  6834  =  669,732     tiplied  by  100,  and  2  times  the  number 

subtracted  from  that  product. 
Find  the  product  of : 

2.  39875x99. 

3.  24567x97. 

4.  14815x98. 

5.  42160x999. 

6.  74853x998. 


7.  412567x99. 

8.  351428x98. 

9.  524167x96. 

10.  674568x997. 

11.  864254x996. 


185.  To  multiply  when  one  part  of  the  multiplier  is  a  factor 
of  another  part. 

1.   Multiply  4256  by  315. 


EXPLANATION.  —  3,  the  number  of  hundreds,  is  a  factor 
of  15,  which  may  be  termed  units.  We  first  multiply 
the  number  by  the  3  hundreds,  and  the  first  figure  of  the 
product  is  written  under  hundreds.  The  15  units  are  5 
times  as  many  units  as  there  are  hundreds,  hence  the 
product  obtained  by  multiplying  by  3  is  multiplied  by  5. 
And  since  the  multiplier  is  regarded  as  units,  the  first  figure  is  written 
in  unit's  place.  The  sum  of  the  partial  products  is  the  entire  product. 


4256 
315 
12768 

63840 
1340640 


Find  the  product  of  : 

2.  3418  x  63. 

3.  4012x93. 

4.  5683  x  279. 

5.  2937x168. 

STAND.    AR.  -  11 


6.  7843  x  213. 

7.  6587x246. 

8.  3826  x  189. 

9.  8834x248. 


162  SHORT  PROCESSES. 

10.  2684x312.  12.   13468x321. 

11.  5168  x  416.  13.  24542  x  328. 

186.  To  multiply  by  the  aliquot  parts  of  100. 

187.  The  parts  of  a  number  which  will  exactly  divide 
it  are  called  the  Aliquot  Parts  of  the  number. 

Thus,  6,  20,  12|,  33i,  etc.,  are  aliquot  parts  of  100. 

The  aliquot  parts  of  100  commonly  used  are  : 
50   =£  of  100        20   =  |  of  100          10  =  ^  of  100 
33^  =  |  of  100        16|  =  i  of  100          8  £  =  -^  of  100 
25   =£oflOO        12i  =  £oflOO          6J  =1*5-  of  100 

Other  parts  of  100  are  : 

40  =  f  of  100  37^  =  f  of  100  66|  =  f  of  100 
60  =  f  of  100  62i  =  |  of  100  75  =  f  of  100 
80  =foflOO  871  =  |  of  100 

1.   Multiply  3429  by 


EXPLANATION.  —  Since  33£  is  £  of  100,  the  number 
*i    oAA     may  ^st  ke  multiplied  by  100,  and  £  of  the  product 
found. 

Multiply  : 

2.  3824  by  25.  7.  8592  by  121  12.  4280  by  75. 

3.  4218  by  50.  8.  9786  by  8J.  13.  6474  by  66|. 

4.  5745  by  20.  9.  14352  by  37£.  14.  8248  by  62f 

5.  6741by33£.  10.  73455  by  33J.  15.  9120  by  87f 

6.  8796byl6f.  11.  94652  by  25.  16.  7560  by  41|. 

188.  To  find  the  cost  when  the  price  by  zoo  or  1000  is 
given. 


WRITTEN  EXERCISES. 

1  .   What  will  385  pounds  of  coal  cost,  at  $  .33  per  hun- 
dred-weight ? 

$    .33  EXPLANATION.  —  Since  100  pounds  cost  $  .  33,  385  pounds,. 

3.85          which  are  equal  to  3.85  times  100  pounds,  will  cost  3.85- 
$  1.2705     times  $  -33,  or  $  1.2705. 


2.  What  will  be  the  cost  of  465  pounds  of  sugar,  at 
$  5.75  per  hundred  pounds  ? 

3.  What  is  the  cost  of  1235  pounds  of  beef,  at  $  6.35 
per  hundred  pounds  ? 

4.  What  must  be  paid  for  1650  pounds  of  coal,  at  $.35 
per  hundred-weight  ? 

5.  What  will  be  the  cost  of  7955  bricks,  at  $  8.75  per  M  ? 

NOTE.  —  The  letters  C  and  M  are  used  instead  of  the  words  hundred 
and  thousand,  respectively. 

6.  When  shingles  are  sold  for  $  5.25  per  M,  how  muck 
must  be  paid  for  8750  ? 

7.  What   will  be  the  cost  of  5268  feet  of  boards,  at 
$31.25  per  M? 

8.  What  must  be  paid  for  a  load  of  hay,  weighing  1592" 
pounds,  when  hay  is  being  sold  for  $7.50  per  ton  (2000 
pounds)  ? 

9.  What  is  the  cost  of  4235  pounds  of  iron,  at  $42.50 
per  ton  ? 

10.  How  much  will  28,750  laths  cost,  at  $2.95  per  M  ? 

11.  What  will  be  the  cost  of  1678  feet  of  pine  boards,  at 
$  19.50  per  M  feet  ? 

12.  How   much  will  15,485  pounds  of  plaster  cost,  at 
$  1.80  per  hundred  pounds  ? 

13.  What  will  375  pineapples  cost,  at  $12.75  per  C  ? 

14.  What  will  960  cocoanuts  cost,  at  $5.45  per  C  ? 


164 


ACCOUNTS  AND   BILLS. 


ACCOUNTS  AND  BILLS. 

189.  The   amount   which,   one   person   owes   another  is 
called  a  Debt. 

190.  The  amount  which  is  due  to  a  person,  or  a  sum 
paid  toward  discharging  a  debt,  is  called  a  Credit. 

191.  A  party  owing  a  debt  is  called  a  Debtor.     A  party 
to  whom  a  debt  is  due  is  called  a  Creditor. 

192.  A  record  of  the  debts  and  credits  between  two  parties 
is  called  an  Account. 

193.  The   difference  between  the  amount  of   debt   and 
credits  is  called  the  Balance  of  an  Account. 

194.  A   statement   of  the   quantity  and  price  of   each 
article,  and  the  value  of  the  whole,  is  called  a  Bill. 

A  bill  is  receipted  when  the  words  Received  Payment  or  Paid  are 
written  at  the  bottom,  and  the  creditor's  name  is  signed  either  by  him- 
self or  by  some  authorized  person. 

195.  The  following  abbreviations  are  in  common  use  : 

@,  At.  Cr.,  Creditor.  Fay't,  Payment, 

%,  Account.  Dr.,  Debtor.  Pd.,  Paid. 

Acc't,  Account.  Doz.,  Dozen.  Per,  By. 

Bal.,  Balance.  Hhd.,  Hogshead.  Rec'd,  Received. 

Bbl.,  Barrel.  Lb.,  Pound.  Yd.,  Yard. 


1. 


RECEIPTED  BILL. 

Chicago,  III.,  July  1,  1892. 


Bought  of   DAVID    C.    BACON. 


tforf* 


t&a,, 


/SO 
70 


88 

05 


07 


WRITTEN  EXERCISES.  165 

196.    Make  out  in  proper  form,  find  the  footings,  and 
receipt  the  following  bills : 

2.  Mrs.  H.  D.  Garmon  bought  of  J.  B.  Hoke  &  Co.,  25 
yards  of  calico  at  7  cents  a  yard,  36  yards  of  muslin  at  & 
cents  a  yard,  and  4  pairs  of  hose  at  $  .40  a  pair. 

3.  Mr.  John  Hood  bought  of  William  Cole  &  Co.,  4  yd. 
of  broadcloth  at  $4.25  a  yard,  12  yd.  of  silk  at  $1.80  a 
yard,  7  yd.  of  flannel  at  $  .45  a  yard,  and  9  yd.  of  lace  at 
$  .35  a  yard. 

4.  Mr.  Henry  Clark  bought  of  Edward  Kill,  16  yd.  of 
cashmere  at  $  1.25  a  yard,  11  yd.  of  cambric  at  10  cents  a. 
yard,  and  18  yd.  of  gingham  at  12^  cents  a  yard. 

5.  Mr.  H.  K.  Martin  bought  of  H.  C.  Allen,  4  pounds  of 
coffee  @  28  cents,  18  pounds  of  sugar  @  5^-  cents,  6  pounds 
of  prunes  @  12^  cents,  2  pounds  of  tea  @  60  cents,  and  3 
pounds  of  rice  at  10  cents. 

6.  Mr.  D.  E.  West  bought  of  Camp  Bros.  &  Co.,  2  spring 
bottom  beds  @  $  12.50,  6  cane  seat  chairs  @  $  1.75,  2  cane- 
seat  rockers  @  $4.50,  3  cottage  bedsteads  @  $7,  and  1 
lounge  @$8.75. 

7.  Mr.  S.  G-.  Eose  bought  of  James  Conrad,  25  yd.  of 
silk  at  $1.80,  8  yd.  of  French  broadcloth  @  $4.75,  14  yd. 
of  Merrimac  prints  @  7  cents,  6  yd.  of  Irish  linen  @  $  .68f , 
and  3  tablecloths  @  $  2.25. 

8.  L.  Eoberts  &  Co.  sold  to  Mrs.  C.  Eoland,  2  doz.  silver 
table  forks  @  $35  a  dozen,  1  doz.  silver  tablespoons  for 
$  25,  2  sets  of  silver  teaspoons  @  $  7.25  a  set,  and  1  silver 
butterdish  for  $6. 

9.  Mr.  Eobert  Homer  bought  of  A.  E.  Young  &  Co.,  4J- 
tons  of  stove  coal  @  $  4.75,  7  tons  of  grate  coal  @  $  5.25, 
and  4  cords  of  wood  @  $  4.60. 


166  REVIEW  EXERCISES. 

10.  Mr.  C.  Dixon  bought  of  H.  White  &  Co.,  18  reams  of 
<3ommercial  note  paper  @  $  1.40,  4500  envelopes  @  $  3.75 
per  M,  10  gross  steel  pens  @  $  .75  per  gross,  50  arithmetics 
<§  $  1.25,  and  20  physiologies  @  $  .90. 

11.  Mr.  David  Brook  bought  of  F.  Taylor,  15  sacks  of  flour 
@  $  .65,  40  pounds  of  Eio  coffee  @  $  .25,  18  Ib.  of  butter 
<@  $  .28,  12  Ib.  of  lard  @  6£  cents,  64  Ib.  of  ham  @  12  cents, 
and  10  Ib.  of  cheese  @  121  cents. 

12.  Mr.  E.  B.  Cooper  bought  of  John  Love,  410  bushels  of 
corn  @  $  .55,  280  bushels  of  wheat  @  $  .98,  175  bushels  of 
oats  @  $  .32,  and  4£  tons  of  hay  @  $  8.75. 

13.  Mr.  J.  D.  Black  bought  of  Baker  &  Leas,  4560  feet  of 
iemlock  @  $13.25  per  M,  9725  feet  of  pine  flooring  @ 
$23.75  per  M,  3560  feet  of  clear  pine  @  $40  per  M,  and 
4275  feet  of  oak  joists  @  $33  per  M. 

14.  Dell  &  Co.  bought  of  Kale  &  Co.,  25  sack  coats  @ 
$  4.25,  48  vests  @  $  1.75,  7  doz.  felt  hats  @  $  27  per  dozen, 
S  doz.  pairs  of  suspenders  @  $  .38  per  pair,  and  4  doz.  pairs 
of  gloves  @  $  .65  per  pair. 

15.  Mr.  H.  K  Biggs  bought  of  K.  E.  Butler,  2  plows  @ 
$11.75,  1  harrow  for  $  9.50,  2  shovels  @  $  1.10,  and  2  steel 
forks  @  $  1.25. 

16.  Mr.  T.  E.  Eanck  bought  of  C.  A.  Hamlin  &  Co.,  45 
yd.  of  tapestry  carpet  @  $  .67^-,  22  yd.  Brussels  carpet  @ 
$  1.90,  and  12f  yd.  of  oilcloth  @  $  .37f 

REVIEW  EXERCISES. 

197.     1.   At  19  cents  a  pound,  how  many  pounds  of  honey 
can  be  bought  for  $  3.99  ? 

2.  If  shovels  are  worth  $  .85  apiece,  how  many  can  be 
bought  for  $  22.10  ? 

3.  A  cubic  inch  of  water  weighs  252.458  grains  avoirdu- 
pois.    How  much  do  231  cubic  inches,  or  a  gallon,  weigh  ? 


REVIEW  EXERCISES.  167 

4.  A  farmer  sold  his  corn  at  $.87J-  per  bushel,  and 
received  for  it  $  131.25.     How  many  bushels  did  ne  sell  ? 

5.  I  bought  3  loads  of  wood,  the  first  containing  1.04 
cords,  the  second  1.05  cords,  and  the  third  .946  cords.     What 
did  it  cost  at  $  3.50  a  cord  ? 

6.  What  will  465  pounds  of  sugar  cost  at  $  4.25  a  hun- 
dred-weight ? 

7.  There  are  2150.42  cubic  inches  in  a  bushel.     How 
many  cubic  inches  are  there  in  10000  bushels  ? 

8.  A  and  B  have  360  acres  of  land,  of  which  A  owns  .37^- 
and  B  .62J.     How  many  acres  has  each  ? 

9.  A  man  sold  .26  of  his  wheat  to  one  man,  and  .39  of  it 
to  another,  and  kept  70  bushels.     How  much  had  he  before 
selling  ? 

10.  What  is  the  product  of  12  x  12  hundred-thousandths  ? 

11.  What  is  the  quotient  when  12  is  divided  by  12  mil- 
lionths  ? 

12.  What  is  the  quotient  when  12  millionths  is  divided 
by  12  thousandths  ? 

13.  How  many  days  must  a  laborer  work  at  $  1.12^-  a  day, 
to  pay  for  6  cords  of  wood  at  $  3.37^  per  cord  ? 

14.  A  farmer  sold  35.5  bushels  of  wheat  at  $  .98  a  bushel, 
and  a  certain  number  of  bushels  of  oats  at  $  .35  a  bushel. 
He  received  for  his  oats  $  17.28  more  than  for  his  wheat. 
How  many  bushels  of  oats  did  he  sell  ? 

15.  If  a  man  can  travel  33.68  miles  in  .8  of  a  day,  how 
far  can  he  travel  in  7.5  days  ? 

16.  I  bought  a  farm  of  71.5  acres  for  $  6220.50.     What 
did  it  cost  me  per  acre  ? 

17.  What  is  the  value  of  297,560  bricks  at  $7.62£  per 
thousand  ? 

18.  An  architect  estimates  that  1,468,000  bricks  will  be 
needed  for  a  school  building.    What  will  they  cost  at  $  7.75 
per  thousand  ? 


168  REVIEW  EXERCISES. 

19.  The  wheel  of  a  bicycle  is  9.13  feet  around.     How 
many  times  will  it  turn  in  going  a  mile,  or  5280  feet  ? 

20.  A  man  owning  .4725  of  a  vessel,  sold  .3  of  his  share. 
What  part  had  he  left  ? 

21.  The  distance  around  a  circle  is  about  3.1416  times 
the  distance  across  it.     If  the  distance   across  a  certain 
circular  race-course  is   1710   feet,  what    is    the  distance 
around  it  ? 

22.  Two  men  start  from  the  same  place  at  the  same  time 
and  travel  in  opposite  directions.     One  goes  4.31  miles  an 
hour,  the  other  3.92  miles  an  hour.     How  far  apart  will  they 
be  in  17  hours  ? 

23.  Eeduce  ^*-  X  ( -  +  - )  to  its  simplest  decimal  form. 

18^      ^9     oj 

24.  A  grocer  bought  15  barrels  of  sugar,  each  containing 
219  pounds,  for  $  125,  and  sold  it  at  5  cents  a  pound.     What 
was  his  gain  ? 

25.  If  .671  of  a  ton  of  hay  is  worth  $  7.50,  what  are  6.75 
tons  worth  ? 

26.  If  the  price  of  gas  is  $1.75  per  thousand  cubic  feet, 
find  the  amount  of  a  man's  bill  when  11,350  cubic  feet  have 
been  consumed. 

27.  At  $57.60  per  acre,  what  are  three  fields  worth  con- 
taining, respectively,  14.6  acres,  20.25  acres,   and  27.625 
acres  ? 

28.  A  real  estate  agent  having  2735  acres  of  land  to  sell, 
sold,  at  different  times,  183.26  acres,  412.625  acres,  640  acres, 
150.875  acres,  240.5  acres,  and  61.971  acres.    How  much 
remained  unsold  ? 

29.  A  merchant  bought  150  barrels  of  apples  for  $1.87^ 
a  barrel.     He  sold  seven  tenths  of  them  at  $1.95  a  barrel, 
and  the  remainder  at  $  1.80  a  barrel.     Did  he  gain  or  lose, 
and  how  much  ? 


REVIEW  EXERCISES.  169 

30.  How  many  rods  of  fence  will  surround  a  rectangular 
field  29.0345  rods  long  and  22.3265  rods  wide  ? 

31.  A  flour  dealer  bought  326  barrels  of  flour  at  $  5.25 
per  barrel.     He  sold  58  barrels  at  a  loss  of  $  .37J  per  barrel, 
How  must  he  sell  the  rest  per  barrel  to  gain  $12  on  the 
investment  ? 

32.  Eeduce  f?i -*- ^  x  -  +  .01  to  a  decimal. 

\4f      41;     9 

33.  What  part  of  A*  :~  3'5 

34.  A  has  13J  cords  of  wood  in  one  pile,  15.66|  cords  in 
a  second,  18J-  cords  in  a  third,  and  21^-  cords  in  a  fourth. 
How  many  cords  has  he  in  all,  and  what  is  the  wood  worth 
at  $  4.25  per  cord  ? 

35.  Twenty-three  miles  of  a  railroad,  47.95  miles  long, 
cost  $  11,578.40  per  mile ;  12  miles  cost  $  13,357.82  per  mile, 
and  the  remainder  cost  $  19,125.26  per  mile.     What  was  the 
average  cost  per  mile  of  the  entire  road  ? 

36.  A  contractor  built  a  house  for  $3575.     The  material 
cost  him  $2150.65,  and  he  employed  15  men  for  6^-  weeks 
of  6  days  each,  at  $2.10  per  day.     Did  he  gain  or  lose 
money,  and  how  much  ? 

37.  What  is  the  value  of  f — - 


2  4  2 

38.  Mr.  D.  H.  Noble  bought  of  Henry  Daron  &  Co.,  9  pairs 
of  calf  boots  at  $  4.25  a  pair,  7  pairs  of  kip  boots  at  $  3.15  a 
pair,  12  pairs  of  ladies'  kid  shoes  at  $  2.65  a  pair,  and  8  pairs 
of  ladies'  cloth  shoes  at  $  2.25  a  pair.    Make  out  and  receipt 
the  bill. 

39.  Mrs.  B.  D.  Boss  bought  of  Cook  &  Co.,  Philadelphia, 
14  yards  of  silk  at  $  1.37^  a  yard,  45  yards  of  sheeting  at 
7  cents  a  yard,  9  handkerchiefs  at  25  cents  apiece,  3  pairs 
of  kid  gloves  at  $  1.12^  a  pair,  and  5  neckties  at  50  cents 
each.    Make  out  and  receipt  this  bill  as  clerk  for  Cook  &  Co. 


DENOMINATE    NUMBERS. 


198.  A  concrete  number  in  which  the  unit  of  measure  is 
established  by  law  or  custom  is  called  a  Denominate  Number. 

Thus,  7  dollars,  2  feet,  4  inches,  5  hours,  8  quarts,  6  pounds,  are 
denominate  numbers. 

199.  A  denominate  number  which  is  composed  of  units  of 
one  denomination  only  is  called  a  Simple  Denominate  Number. 

Thus,  5  ounces,  7  yards,  3  miles,  6  hours,  10  pounds,  12  quarts,  are 
simple  denominate  numbers. 

200.  A  denominate  number  which  is  composed  of  units 
of  two  or  more  denominations  that  are  related  to  each  other, 
is  called  a  Compound  Denominate  Number. 

Thus,  3  yards,  2  feet,  4  inches,  is  a  compound  denominate  number. 
So  also  is  1  year,  5  months,  3  days. 

201.  A  unit  of  measure,  from  which  other  units  of  the 
same  kind  may  be  derived,  is  called  a  Standard  Unit. 

Thus,  the  yard  is  the  standard  unit  of  length,  because  the  other  units 
are  derived  from  it. 

202.  The  ratio  by  which  numbers  increase  and  decrease 
is  called  a  Scale. 

Scales  are  either  uniform  or  varying. 

Thus,  in  United  States  currency  the  scale  is  uniform,  being  decimal ; 
in  Linear  measure  it  is  varying,  for  12  inches  equal  one  foot,  3  feet  one 
yard,  etc. 

170 


REDUCTION.  171 

REDUCTION. 
LINEAR   MEASURES. 

203.  That  which  has  length  only  is  called  a  Line. 
Thus,  the  distance  between  two  objects  or  places  is  a  line. 

204.  Measures  that  are  used  in  measuring  length  only  are 
called  Linear  Measures. 

NOTE.  —  The  tables  of  Denominate  Numbers  will  be  found  on  page 
418  and  the  subsequent  pages. 

1.  How  many  inches  are  there  in  5  ft.  ?    In  7  ft.  ? 

2.  How  many  feet  are  there  in  5  yd.  ?     In  7  yd.  ? 

3.  How  many  yards  are  there  in  2  rd.  ?     In  4  rd.  ? 

4.  How  many  feet  are  there  in  2  rd.  ?     In  4  rd.  ? 

5.  How  many  rods  are  there  in  2  mi.  ?    In  3  mi.  ?    In 
10  mi.  ?     In  30  mi.  ?     In  100  mi.  ? 

6.  How  many  inches  are  there  in  2  ft.  6  in.  ?    In  3  ft. 
4  in.  ?    In  5  ft.  ?    In  5  ft.  2  in.  ? 

7.  How  many  feet  are  there  in  2  yd.  2  ft.  ?    In  3  yd. 
2  ft.  ?     In  10  yd.  ?     In  10  rd.  ? 

8.  How  many  yards  are  there  in  2  rd.  3  yd.  ?    In  4  rd. 
2  yd.  ?    In  10  rd.  2  yd.  ?     In  1  mi.  ? 

9.  How  many  rods  are  there  in  1  mi.  80  rd.  ?     In  2  mi. 
60  rd.  ?     In  5  mi.  80  rd.  ? 

10.   How  many  inches  are  there  in  -J  ft.  ?    In  J  ft.  ?    In 
Jffc.?.  In  f  ft.?     In  |  ft.  ? 

205.  The  process  of  changing  a  denominate  number  from 
one  denomination  to  another  without  altering  its  value  is 
called  Reduction. 

206.  The  process  of  changing  a  denominate  number  to 
an  equivalent  number  of  a  lower  denomination  is  called 
Reduction  to  Lower  Denominations  or  Reduction  Descending. 


172  DENOMINATE  NUMBERS. 

WRITTEN   EXERCISES, 
o 

o 

Tg         i.   Eeduce  5  yd.  2  ft.  8  in.  to  inches. 

EXPLANATION.  —  Since  there  are  3  feet  in  1  yard,  in  5  yards 

17     there  are  5  times  3  feet  =  15  feet,  and  15  feet  +  2  feet  =  17  feet. 

12          Since  there  are  12  inches  in  1  foot,  in  17  feet  there  are  17  times 

OAT     12  inches  =  204  inches,  and  204  inches  -f  8  inches  =  212  inches. 

g          Hence  5  yd.  2  ft.  8  in.  =  212  inches. 

212 

RULE.  —  Multiply  the  number  of  the  highest  denomination 
given,  by  the  number  indicating  how  many  units  of  the  next 
lower  denomination  are  equal  to  one  of  the  higher,  and  to  the 
product  add  the  number  given  of  this  lower  denomination. 

Proceed  in  like  manner  with  this  and  each  successive  result 
thus  obtained,  until  the  number  is  reduced  to  the  required 
denomination. 

Eeduce  to  feet : 

2.  4  rd.  2  yd.  2  ft.  6.   30  rd.  6  ft. 

3.  6  rd.  3  yd.  1  ft.  7.    2  mi.  15  rd.  8  ft. 

4.  5  rd.  4  yd.  2  ft.  8.    3  mi.  25  rd.  12  ft. 

5.  13  rd.  5  yd.  2  ft.  9.   5  mi.  100  rd.  15  ft. 
Eeduce  to  inches : 

10.  2  yd.  2  ft.  2  in.  13.  25  rd.  12  ft.  4  in. 

11.  3  yd.  1  ft.  4  in.  14.   3  mi.  40  rd.  8  ft.  7  in. 

12.  5  yd.  2  ft.  6  in.  15.   5  mi.  50  rd.  5  yd.  3  ft.  4  in, 

16.  Eeduce  ^  of  a  rod  to  units  of  lower  denominations. 

SOLUTION. 

f  of  a  rod  =  f  of  V  yd.  =  f  f  yd.  =  2&  yd. 
&  of  a  yd.  =  T\  of  3    ft.  =  |f  ft.  =  1^  ft. 
•fa  of  a  ft.  =  TV  of  12  in.  =  if  in. 
.-.  f  of  a  rod  =  2  yd.  1  ft.  £f  in. 

Eeduce  to  units  of  lower  denominations : 

17.  -f  rd.  19.     f  rd.  21.    £  mi.  23.     $  mi. 

18.  f  rd.  20.    ^rd.  22.    f  mi.  24.    ^  mi. 


REDUCTION.  173 

25.  Keduce  .885  of  a  yd.  to  feet  and  inches. 

SOLUTION. 

.885  of  a  yd.  =  .885  of    3  ft.  =  2.655  ft. 
.655  of  a  ft.    =  .655  of  12  in.  =  7.860  in. 
.-.   .885  of  a  yd.  =  2  ft.  7.86  in. 

Keduce  to  units  of  lower  denominations  :  . 

26.  .75  yd.      28.    .625  yd.      30.    .375  rd.      32.    .725  mi. 

27.  .95  yd.      29.    .875  yd.      31.    .645  rd.      33.    .975  mi. 

207.  1.   How  many  feet  are  there  in  48  in.  ?     In  72  in.  ? 

2.  How  many  yards  are  there  in  30  ft.  ?     In  48  ft.  ? 

3.  How  many  rods  are  there  in  11  yd.?     In  22  yd.? 

4.  How  many  rods  are  there  in  33  ft.  ?    In  66  ft.  ? 

5.  How  many  miles  are  there  in  640  rd.  ?    In  960  rd.  ? 

6.  What  part  of  a  rod  are  8J  ft.  ?    4£  ft.  ?     1|  ft.  ? 

208.  The  process  of  changing  a  denominate  number  to 
an  equivalent  number  of  a  higher  denomination  is  called 
Reduction  to  Higher  Denominations  or  Reduction  Ascending. 

WRITTEN  EXERCISES. 
1.   Reduce  641558  in.  to  miles,  etc. 


12 

3 


11 
320 


641558  in.  EXPLANATION.  —  Since  there 


53463  ft    4-  2  in  are  12  incnes  in  1  ft.,  in  641558 


QO-I     A       c\  f  inches  there  are  as  many  feet  as 

17821  yd.  +  0  It.            .  12  m>  are  contained  times  in 

£  641558  in. ,  or  53463  ft.  and  2  in. 

35642                     [or  1  yd.  Since  there  are  3  ft.  in  1  yd., 


3240  rd.  +2  half-yards,  m  53463  f>  there  are 
TA      •        A  f\    A  yards  as  3  ft.  are  contained  times 

•  ^A  "t.  V  A  o  •    in  53463  ft" or  17821  yd- 

Ans.  10  mi.  40  rd.  1  yd.  2  in.  since  there  are  5|  yd.  in  1  rd., 
in  17821  yd.  there  are  as  many  rods  as  5|  yd.  are  contained  times  in 
17821  yd.,  or  what  is  the  same  thing,  as  many  times  as  11  half-yards 
are  contained  times  in  35642  half-yards,  or  3240  rd.  and  2  half-yards, 
or  1  yd.  remaining. 

Since  there  are  320  rd.  in  1  mi. ,  in  3240  rd.  there  are  as  many  miles 
as  320  rd.  are  contained  times  in  3240  rd.,  or  10  mi.  and  40  rd. 

.-.  641568  in.  =  10  mi.  40  rd.  1  yd.  2  in. 


174  DENOMINATE  NUMBERS. 

RULE.  —  Divide  the  given  number  by  the  number  indicating 
how  many  units  of  the  given  denomination  make  one  of  the 
next  higher  denomination. 

Proceed  in  like  manner  with  this,  and  each  successive  quo- 
tient, till  the  whole  is  reduced  to  the  required  denomination. 

The  last  quotient,  with  the  remainders,  if  any,  annexed,  will 
be  the  required  ansiver. 

Reduce  to  higher  denominations  : 

2.  1320  in.  7.    88792  in.  12.  99999  yd. 

3.  4254  in.  8.    96450  in.  13.  425644  in. 

4.  7560ft.  9.   75680ft.  14.  586100  in. 

5.  16890yd.  10.    97480yd.  15.  76840ft. 

6.  42560yd.  11.    98764ft.  16.  876400  in. 

17.   Eeduce  ^-  of  a  ft.  to  the  fraction  of  a  mile. 

SOLUTION. 

1  ft.    =  i  of  a  yd.  .-.  A  ft-     =  TV  of  I  yd.       =  &  yd- 

1  yd.  =  fV  Of  a  rd.         ...  £  yd.   =  ^  of  A  rd.      =  fa  rd. 
1  rd.  =  fa  of  a  mi.       .-.  fa  rd.  =  fa  of  fa  mi.  =  ^fa  mi. 

Or, 
1  ft-    =  «W  of  a  mi-     •'•  T5<r  ft-     =  T\  of  **W  mi-  =  rainr  mi. 


Reduce  to  the  fraction  of  a  rod  : 

18.  |  ft.  20.  .635  ft.  22.  ^  in. 

19.  f  ft.  21.  IJ-in.  23.  .375  in. 

Reduce  to  the  fraction  of  a  mile  : 

24.  -f  rd.  26.   .44  rd.  28.  ^  ft. 

25.  ^  rd.  27.  f  ft.  29.  .35  ft. 


SURFACE  MEASURES. 


175 


12 


30.  Keduce  3  yd.  2  ft.  6  in.  to  the  decimal  of  a  rod. 

EXPLANATION. — Since  there  are  12  in.  in  1  ft.,  ^ 
of  the  number  of  inches  equals  the  number  of  feet. 
y1^  of  6  equals  .5 ;  therefore  there  are  2.5  ft.  Since 
there  are  3  ft.  in  1  yd.,  J  of  the  number  of  feet 
equals  the  number  of  yards.  J  of  2.5  is  .8333  +  ; 
therefore  there  are  3.8333  +  yd.,  etc. 


2.5 


in. 
ft. 


3.8333+  yd. 


.6969+  rd. 


Reduce  to  the  decimal  of  a  rod : 
31.  3  yd.  2  ft.  8  in.     32.  4  yd.  1  ft.  6  in.     33.  2  yd.  2  ft.  5  in. 

Express  as  rods  and  decimals  of  a  rod : 
34.  4  rd.  3  yd.  1  ft.  5  in.  35.  8  rd.  1  yd.  2  ft.  9  in. 


SURFACE    MEASURES. 

209.  Anything  that  has  only  length  and  breadth  is  called 
a  Surface. 

Thus,  this  page,  the  floor,  or  the  outside  of  anything  is  a  surface. 

210.  The   difference  in  the  direction  of  two  lines  that 
meet  is  called  an  Angle. 


211.  A  figure  that  is  bounded  by  four 
equal  straight  sides  and  has  four  equal 
angles  is  called  a  Square. 

A  square  inch  is  a  square  each  of  whose 
sides  is  one  inch  long ;  a  square  foot  is  a  square 
each  of  whose  sides  is  one  foot  long. 

The  angles  of  a  square  are  called  right 
angles. 


A  figure  that  has  four  straight 
sides  and  four  right  angles  is  called  a 
Rectangle. 

It  will  be  seen  that  a  square  is  a 
rectangle  whose  four  sides  are  equal 
each  to  each. 


ANOLB. 


SQUARE. 


RECTANGLE. 


176 


DENOMINATE  NUMBERS. 


213.  The  number  of  square  units  in 
the  surface  of  anything  is  called  its 
Area. 

Thus,  if  a  rectangle  is  4  inches  long  and  3 
inches  wide,  the  area  will  be  12  square  inches. 

For  it  may  be  divided  into  4  rows,  each  con- 
taining 3  square  inches  or  units,  and  the  entire 
area  will  be  12  square  inches. 


214.   PRINCIPLE.  —  TJie  area  of  a  rec- 
tangle is  equal  to  the  product  of  the  numbers  that  express  its 
length  and  breadth. 

The  length  and  breadth  must  be  expressed  in  units  of  the  same 
denomination ;  the  product  will  then  express  the  area  in  square  units 
of  the  same  name. 

1.  How  many  square  inches  are  there  in  the  surface  of 
a  rectangle  that  is  6  in.  long  and  4  in.  wide  ?     In  one  8  in. 
long  and  5  in.  wide  ?     In  one  7  in.  long  and  6  in.  wide  ? 

2.  How  many  square  feet  are  there  in  the  surface  of  a 
rectangle  9  ft.  by  5  ft.  ?     In  one  8  ft.  by  7  ft.  ?     In  one  10  ft. 
by  8ft? 

3.  How  many  square  inches  are  there  in  a  square  whose 
sides  are  5  in.?    6   in.?    8  in.?    12  in.,  or  1  ft.?     How 
many  square  inches,  then,  are  there  in  a  square  foot  ? 

4.  How  many  square  feet  are  there  in  a  square  whose 
sides  are  2  ft.  ?  3  ft.,  or  1  yd.  ?     How  many  square  feet  are 
there,  then,  in  a  square  yard  ? 

5.  How  many  square  yards  are  there  in  a  square  whose 
sides  are  3  yd.  ?   4  yd.  ?   5  yd.  ?    5 J  yd.,  or  a  rod  ?     How 
many  square  yards  are  there  in  a  square  rod  ? 

6.  How  many  square  rods  are  there  in  a  rectangle  8  rd. 
by  6  rd.  ?  10  rd.  by  12  rd.  ?  10  rd.  by  16  rd.  ?    A  rectangle 
that  contains  160  square  rods  is  called  an  Acre. 


SURFACE  MEASURES.  17T 

7.  Huw  many  rods   are   there   in  a  mile  ?     How  many- 
square  rods  are  there  in  a  square  whose  sides  are  each  a. 
mile,  or  how  many  square  rods  are  there  in  a  sq.  mi.  ? 

8.  Since  there  are  160  sq.  rd.  in  an  acre,  how  many 
acres  are  there  in  a  square  mile  ? 

9.  Write  out  a  table  of  square  measures. 

WRITTEN   EXERCISES. 
Eeduce  to  square  inches  : 

1.  4  sq.  yd.  5  sq.  ft.  6.  120  sq.  rd.  120  sq.  in. 

2.  9  sq.  yd.  3  sq.  ft.  7.  5  A.  20  sq.  yd. 

3.  20  sq.  yd.  4  sq.  ft.  8.  8  A.  45  sq.  rd. 

4.  8  sq.  rd.  4  sq.  yd.  9.  30  A.  5  sq.  rd. 

5.  12  sq.  rd.  7  sq.  ft.  10.  2  sq.  mi.  80  sq.  rd. 

Eeduce  to  higher  denominations  : 

11.  7460  sq.  in.        14.  12340  sq.  ft.  17.  102400  sq.  rd. 

12.  6720  sq.  in.        15.  7580  sq.  yd.  18.  387690  sq.  rd. 

13.  8000  sq.  in.        16.  9678  sq.  yd.  19.  3968479  sq.  ft. 

Eeduce  to  units  of  lower  denominations  : 

20.  |  sq.  rd.    22.  -f-  sq.  rd.        24.  ^  sq.  yd.      26.  ^  sq.  rd.. 

21.  .625  A.      23.  .545  sq.  rd.    25.  .875  sq,  ft.    27.  .495  A. 

28.  Eeduce  ^  of  a  sq.  yd.  to  the  fraction  of  a  sq.  rd. 
SOLUTION.         1  sq.  rd.  =  301,  or  if i  sq.  yd. 

1  sq.  yd.  =  Tf  T  sq.  rd. 

f  sq.  yd.  =  f  of  Tf T  sq.  rd.  =  ffi  sq.  rd. 

29.  Eeduce  |-  of  a  sq.  ft.  to  the  fraction  of  a  sq.  rd. 

30.  Eeduce  -|  of  a  sq.  in.  to  the  fraction  of  a  sq.  yd. 

31.  Eeduce  .65  of  a  sq.  yd.  to  the  fraction  of  an  A. 

32.  Eeduce  5  sq.  ft.  100  sq.  in.  to  the  decimal  of  a  sq.  yd~ 

STAND.  AR. —  12 


178 


DENOMINATE   NUMBERS. 


MEASURES  OF  VOLUME. 

215.  Anything  which  has  length,  breadth,  and  thickness 
is  called  a  Solid. 

216.  A  solid  having  six  equal  square  sides  or  faces  is 
called  a  Cube. 

A  cube  whose  sides  or  faces  are  each  a 
square  inch  is  called  a  Cubic  Inch.  One 
whose  sides  are  each  a  square  foot  is 
called  a  Cubic  Foot.  One  whose  sides 
are  each  a  square  yard  is  called  a  Cubic 
Yard. 

217.  The  number  of  solid  units  any  body  contains  is  its 
Solid  Contents  or  Volume. 

Thus,  if  a  solid  is  4  ft.  long,  3  ft.  wide, 
and  3  ft.  thick,  its  solid  contents  or  vol- 
ume is  36  cubic  feet.  For  it  may  be 
divided  into  3  blocks,  each  containing 
12  cubic  feet.  The  number  of  cubic 
feet  in  each  block  is  equal  to  the  product 
of  the  numbers  expressing  the  length 
and  breadth  of  the  solid,  and  the  number 
of  blocks  is  equal  to  the  number  of 
units  of  thickness.  Therefore, 

218.  PRINCIPLE.  —  The  volume  of  any  rectangular  solid  is 
equal  to  the  product  of  the  numbers   expressing  its  length, 
breadth,  and  thickness. 

The  length,  breadth,  and  thickness  must  be  expressed  in  units  of 
the  same  denomination  ;  the  product  will  then  express  the  volume  in 
cubic  units  of  the  same  name. 

1.  How  many  cubic  inches  are  there  in  a  rectangular 
solid  3  in.  long,  2  in.  wide,  and  2  in.  thick  ? 

2.  How  many  cubic  inches  are  there  in  a  rectangular 
solid  1  ft.  long,  1  ft.  wide,  and  1  ft.  thick,  or  a  cubic  foot  ? 


MEASURES   OF   VOLUME. 


179 


3.  How  many  cubic 
feet  are  there  in  a  cube 
whose  edge  is  1  yd.  ? 

4.  A  cord  of  wood  is 
8  ft.  long,  4  ft.  wide,  and 
4  ft.  high.     How  many 
cubic  feet  does  it  con- 
tain? 

5.  Write  a  table  of  cubic  measures. 


5.  80  cu.  yd.  16  cu.  ft. 

6.  2  C.  8  cu.  ft. 

7.  5  C.  13  cu.  ft. 

8.  15  cu.  ft.  1115  cu.  in. 


WRITTEN    EXERCISES. 
Reduce  to  cubic  inches : 

1.  15  cu.  ft.  120  cu.  in. 

2.  32  cu.  ft.  114  cu.  in. 

3.  40  cu.  yd.  18  cu.  ft. 

4.  60  cu.  yd.  25  cu.  ft. 

Reduce  to  units  of  higher  denominations : 
9.    148760  cu.  in.  12.   724570  cu.  in. 

10.  96780  cu.  in.  13.   426790  cu.  in. 

11.  69875  cu.  in.  14.    6037860  cu.  in. 

15.  Reduce  -J-  of  a  cu.  yd.  to  units  of  lower  denominations. 

16.  Reduce  £  of  a  cu.  ft.  to  cubic  inches. 

17.  Reduce  J  of  a  cu.  yd.  to  cubic  feet  and  inches. 

18.  Reduce  .675  of  a  cu.  ft.  to  cubic  inches. 

19.  Reduce  £  of  a  cu.  ft.  to  a  fraction  of  a  cubic  yard. 

20.  Reduce  8  cu.  ft.  240  cu.  in.  to  the  decimal  of  a  cubic 
yard. 

21.  A  man  bought  a  block  of  marble  4  ft.  9  in.  long,  2  ft. 
7  in.  wide,  and  2  ft.  5|  in.  thick.     How  much  did  he  pay 
for  it  at  the  rate  of  $  15.80  per  cubic  yard  ? 


180  DENOMINATE   NUMBERS. 

SURVEYORS'  MEASURES. 
SURVEYORS'  LINEAR  MEASURE. 

219,    Surveyors,  in  measuring,  use  a  chain  consisting  of 
100  links,  its  length  being  4  rods,  or  66  feet. 

1.  Since  a  chain  contains  a  hundred  links,  how  many 
links  make  a  rod  ? 

2.  How  many  rods  make  a  mile  ?     How  many  chains  ? 

3.  How  many  inches  are  there  in  a  chain,  or  66  ft.  ? 

4.  Since  there  are  100  links  in  a  chain,  what  is  the 
length  of  each  link  ? 

5.  Write  a  table  of  surveyors'  linear  measures. 

6.  How  many  links  in  2  rd.  ?     In  3  rd.  ?     In  4  rd.  ? 

7.  How  many  ch.  in  20  rd.  ?     In  32  rd.  ?     In  40  rd.  ? 

WRITTEN  EXERCISES. 
Beduce  to  links : 

1.  40  rd.  151.  5.   25  ch.  3  rd.  20  1. 

2.  3  rd.  12|  1.  6.    40  ch.  60  1. 

3.  5rd.  151.  7.   75  ch.  751. 

4.  7  ch.  2  rd.  14  1.  8.    12  mi.  16  ch.  20  1. 
Keduce  to  units  of  lower  denominations  : 

9.    |  ch.  11.    I  rd.  13.  .675  mi. 

10.    -f-  mi.  12.    |  ch.  14.  .595  ch. 

Reduce  to  units  of  higher  denominations : 

15.  792  in.  17.    76851.  19.  76489  in. 

16.  876  1.  18.    8436  1.  20.  123456  in. 

21.  Beduce  f  of  a  chain  to  the  fraction  of  a  mile. 

22.  Eeduce  -f  of  a  chain  to  links. 

23.  Eeduce  28  links  to  a  fraction  of  a  chain. 


MEASURES  OF  CAPACITY.  181 

SURVEYORS'  SQUARE  MEASURE. 

220.   1.   How  many  links  are  there  in  a  rod  ?     How  many 
square  links  are  there  in  a  square  rod  ? 

2.  How  many  rods  are  there  in  a  chain?    How  many 
square  rods  are  there  in  a  square  chain? 

3.  How  many  rods  are  there  in  an  acre  ?     How  many 
square  chains  are  there  in  an  acre  ? 

4.  Write  a  table  of  surveyors'  square  measures. 

WRITTEN   EXERCISES. 
Eeduce  to  units  of  lower  denominations : 

1.  5  sq.  ch.  4.    240  acres  15  sq.  ch. 

2.  2  acres  8  sq.  ch.  5.3  sq.  mi.  18  A.  15  sq.  ch. 

3.  4  of  a  chain.  6.    .645  of  an  acre. 

o 


O.        -5-    UJL    <\t    U 110/111.  U»        .V*lt/    U.L    CULL    OftJAWi 

o 

Reduce  to  units  of  higher  denominations : 

7.  890  sq.  ch.         10.    1920  sq.  ch.         13.  3375  sq. 

8.  960  sq.  rd.    11.  2430  sq.  1.     14.  7500  sq 

9.  1875  sq.  1.    12.  3000  sq.  ch.    15.  8640  sq 


1. 

[.rd. 
.1. 


MEASURES  OF  CAPACITY. 
LIQUID  MEASURE. 

221.   1.    How  many  gills  are  there  in  7  pt.  ?     In  10  pt.  ? 

2.  How  many  gills  are  there  in  5  qt.  ?     In  8  qt.  ? 

3.  How  many  pints  are  there  .in  8  qt.  ?    In  10  qt.  ? 

4.  How  many  quarts  are  there  in  50  pt.  ?     In  68  pt.  ? 

WRITTEN  EXERCISES. 
Eeduce  to  units  of  lower  denominations : 

1.  25  gal.  3  qt.  1  pt.  5.   15  gal.  3  qt.  1  pt.  2  gi. 

2.  27  gal.  2  qt.  1  pt.  6.   45  gal.  2  qt.  1  pt.  3  gi. 

3.  30  gal.  3  qt.  2  gi.  7.    .375  gal. 

4.  f  gal.  8.    f  gal. 


182  DENOMINATE   NUMBERS. 

Reduce  to  units  of  higher  denominations : 
9.    196  pt.  12.    1980  gi.  15.    14620  gi. 

10.  428  gi.  13.    1286  qt.  16.    15408  pt. 

11.  680  qt.  14.    1648  pt.  17.    25600  gi. 

18.  Eeduce  -^  of  a  pint  to  the  fraction  of  a  gallon. 

19.  Eeduce  -f-  of  a  gill  to  the  fraction  of  a  gallon. 

20.  Keduce  1  pt.  2  gi.  to  the  decimal  of  a  gallon. 

DRY  MEASURE. 
222.    1.    How  many  pints  are  there  in  10  qt.  ?    In  12  qt.  ? 

2.  How  many  quarts  are  there  in  8  pk.  ?     In  10  pk.  ? 

3.  How  many  quarts  are  there  in  a  bushel  ?    How  many 
pints  ?    In  3  bushels  how  many  quarts  ?    How  many  pints  ? 

4.  How  many  pecks  are  there  in  33  qt.  ?     In  41  qt.  ? 

5.  How  many  pecks  are  there  in  18  pt.  ?    In  24  pt.  ? 

6.  How  many  bushels  are  there  in  20  pk.  ?    In  25  pk.  ? 

WRITTEN  EXERCISES. 

Eeduce  to  pints : 

1.  3  pk.  6  qt.  1  pt.  5.    2  bu.  3  pk.  5  qt.  1  pt. 

2.  5  pk.  5  qt.  1  pt.  6.    6  bu.  2  pk.  3  qt.  1  pt. 

3.  4  bu.  3  pk.  5  qt.  7.    15  bu.  3  qt.  1£  pt. 

4.  fbu.  8.    .625  pk. 

Eeduce  to  units  of  higher  denominations : 

9.    2144  qt.  12.   4640  pt.  15.  3930  gi. 

10.  3360  qt.  13.    3760  pt.  16.  7870  pt. 

11.  5500  qt.  14.   4800  qt.  17.  8000  gi. 

18.  Eeduce  ^-  of  a  pint  to  the  fraction  of  a  bushel. 

19.  Eeduce  -f-  of  a  quart  to  the  fraction  of  a  bushel. 

20.  Eeduce  3  pk.  7  qt.  to  the  decimal  of  a  bushel. 


MEASURES  OF  WEIGHT.  183 

MEASURES  OF  WEIGHT. 

223.  The  measure  of  the  force  which  attracts  bodies  to 
the  earth  is  called  Weight. 

AVOIKDUPOIS  WEIGHT. 

224.  1.    How  many  ounces  are  there  in  5  Ib.  ?    In  8  Ib.  ? 

2.  How  many  pounds  are  there  in  6  cwt.  ?     In  8  cwt.  ? 

3.  How  many  pounds  are  there  in  3  tons  ?     In  5  tons  ? 

4.  How  many  cwt.  are  there  in  5 tons  3  hundredweight? 

5.  How  many  pounds  are  there  in  6  T.  8  cwt.  8  Ib.  ? 

6.  How  many  pounds  are  there  in  48  oz.  ?     In  64  oz.  ? 

7.  How  many  tons  are  there  in  60  cwt.  ?    In  100  cwt.  ? 

WRITTEN   EXERCISES. 

Eeduce  to  units  of  lower  denominations : 

1.  4  cwt.  25  Ib.  12  oz.  5.7  cwt.  16  Ib.  4  oz. 

2.  3  cwt.  76  Ib.  8  oz.  6.    3  T.  15  cwt.  8  Ib.  2  oz. 

3.  2  cwt.  18  Ib.  9  oz.  7.    5  T.  10  cwt.  24  Ib.  8  oz. 

4.  |  of  a  ton.  8.    .675  of  a  cwt. 

Eeduce  to  units  of  higher  denominations : 

9.   1400  oz.  12.   4260  Ib.  15.  14784  oz. 

10.  1056  oz.  13.    7525  Ib.  16.  36450  Ib. 

11.  2080  oz.  14.   8123  Ib.  17.  987696  oz. 

18.  Eeduce  -f-  of  an  ounce  to  the  fraction  of  a  hundred- 
weight. 

19.  Eeduce  f  of  a  pound  to  the  fraction  of  a  ton. 

20.  Eeduce  5  Ib.  8  oz.  to  the  decimal  of  a  ton. 

21.  Eeduce  .725  of  a  pound  to  the  fraction  of  a  ton. 


184  DENOMINATE   NUMBERS. 

TROY  WEIGHT. 

225.   1.   How  many  grains  are  there  in  2  pwt.  ?     In  3 
pwt.  ?    In  2  pwt.  10  gr.  ?    In  3  pwt.  5  gr.  ?    In  4  pwt.  4  gr.  ? 

2.  How  many  pennyweights  are  there  in  4  oz.  ?     In  5 
oz.  ?     In  6  oz.  ?     In  5  oz.  6  pwt.  ?    In  4  oz.  10  pwt.  ? 

3.  How  many  ounces  are  there  in  2  Ib.  ?    In  3  Ib.  ?     In 
3  Ib.   2  oz.  ?     In  5  Ib.  8  oz.  ?     In  10  Ib.  7  oz.  ? 

4.  How  many  ounces  are  there  in  1  Ib.  ?      How  many 
pennyweights  ?     How  many  grains  ? 

5.  How  many  ounces  are  there  in  f  Ib.  ?     In  f  Ib.  ? 

6.  How  many  ounces   and   pennyweights  are  there  in 
|  Ib.  ?     In  |  Ib.  ? 

7.  How  many  pounds  are  there  in  48  oz.  ?     In  72  oz.  ? 

WRITTEN   EXERCISES. 

Reduce  to  units  of  lower  denominations  : 

1.  3  oz.  18  pwt.  20  gr.          5.    8  Ib.  9  oz.  15  pwt.  18  gr. 

2.  7  oz.  12  pwt.  10  gr.          6.    45  Ib.  7  oz.  13  pwt.  15  gr. 

3.  10  oz.  16  pwt.  12  gr.        7.   .875  of  a  Ib. 

4.  f  of  a  pound.  8.    ^-  of  an  ounce. 

Reduce  to  units  of  higher  denominations : 

9.    1940  gr.  12.    4460  pwt.  15.  7896  oz. 

10.  2560  gr.  13.    14520  gr.  16.  9678  pwt. 

11.  1276  pwt.  14.   24676  pwt.         17.  34560  gr. 

18.  Eeduce  |  of  a  pwt.  to  the  fraction  of  a  pound. 

19.  Eeduce  .395  of  a  grain  to  the  fraction  of  an  ounce. 

20.  Eeduce  3  oz.  5  pwt.  to  the  decimal  of  a  pound. 


APOTHECARIES'   WEIGHT.  185 


APOTHECARIES7    WEIGHT. 

226.  1.    How  many  grains  are  there  in  3  3  ?     In  5  3  ? 

2.  How  many  scruples  are  there  in  8  3  ?     In  10  3  1  3  ? 

3.  How  many  drams  are  there  in  53?     In  83  63? 

4.  How  many  drams  are  there  in  1  Ib  ?     In  2  Ib  ? 

5.  How  many  ounces  are  there  in  4  Ib  ?     In  8  Ib  4  3  ? 

6.  How  many  drams  are  there  in  18  3  ?     In  30  3  ? 

WRITTEN   EXERCISES. 
Reduce  to  grains : 

1.  73  23  15  gr.  3.   151b532343. 

2.  655323.  4.    24ft>63  23  15gr. 

Eeduce  to  units  of  higher  denominations  : 

5.  12393-  7.   36483-  9.    12560  gr. 

6.  42603.  8.   82603.  10.    92375  gr. 

227.  Comparison  of  common  weights  and  measures. 

1  pound  Troy  or  Apothecaries'  weight  contains  .     .     .  5760  gr. 

1  pound  Avoirdupois  weight  contains 7000  gr. 

A  bushel  (32  qt.)  contains 2150.4  cu.  hi. 

A  quart  dry  measure  contains 67£  cu.  in. 

A  gallon  liquid  measure  contains 231  cu.  in. 

A  quart  liquid  measure  contains' 57  f  cu.  in. 

1.  A  druggist  bought  5  pounds  of  opium  by  Avoirdupois 
weight  at  $  8  a  pound,  and  sold  it  by  Apothecaries7  weight 
at  $  1  per  ounce.     How  much  did  he  gain  ? 

2.  A  miner  sold  to  a  broker  2  pounds  of  gold  dust  at 
$  220  per  pound  Avoirdupois,  and  the  broker  sold  it  at  $  16 
per  ounce  Troy.     Did  he  gain  or  lose,  and  how  much  ? 

3.  What  part  of  a  pound  Avoirdupois  is  a  pound  Troy  ? 

4.  A  boy  buys  chestnuts  at  $  1.60  per  bu.,  and  sells  them 
at  $  .10  per  quart  liquid  measure.     How  much  is  his  gain 
per  bushel  ? 


186  DENOMINATE  NUMBERS. 

MEASURES  OF  TIME. 

228.   1.    How  many  seconds  are  there  in  5  min.  ?     In  6 
min.  ?     In  8  min.  ?     In  10  min.  20  sec.  ?     In  5  min.  35  sec.  ? 

2.  How  many  minutes  are  there  in  3  hr.  ?     In  5  hr.  ? 
In  6  hr.  ?     In  6  hr.  30  min.  ?     In  8  hr.  20  min.  ?     In  |  hr.  ? 

3.  How  many  hours  are  there  in  2  da.  ?     In  3  da.  ?     In 

5  da.  ?     In  2  da.  5  hr.  ?    In  3  da.  10  hr.  ?     In  |  da.  ? 

4.  How  many  days  are  there  in  5  wk.  ?     In  8  wk.  ?    In 

6  wk.  ?     In  6  wk.  5  da.  ?     In  5  wk.  5  da.  ?     In  f  wk.  ?    In 
f  wk.  ?     In  |  wk.  ? 

5.  How  many  days  are  there  in  two  calendar  years  ?     In 
J  yr.  ?     In  2  leap  years  ?     In  £  of  a  leap  year  ?     What 
years  are  leap  years  ? 

6.  How  many  minutes  are  there  in  120  sec.  ?   In  180  sec.  ? 

7.  How   many  hours   are  there  in  120  min.  ?     In   240 
min.  ?     In  140  min.  ?     In  190  min.  ?     In  270  min.  ? 

8.  How  many  days   are  there  in  48  hr.  ?     In  72  hr.  ? 
In  96  hr.  ?     In  56  hr.  ?     In  80  hr.  ?     In  100  hr.  ? 

WRITTEN   EXERCISES. 
Reduce  to  units  of  lower  denominations  : 

1.  5  hr.  15  min.  12  sec.  5.  4  wk.  2  da. 

2.  6  hr.  27  min.  38  sec.  6.  2  wk.  12  hr. 

3.  2  wk.  5  da.  13  hr.  7.  5  wk.  6  da.  10  hr. 

4.  £  of  a  day.  8.  .785  of  a  day. 

Reduce  to  units  of  higher  denominations  : 
9.  1460  min.  12.  8000  hr.  15.  486950  sec. 

10.  3648  hr.  13.  12000  hr.  16.  867896  sec. 

11.  7432  hr.  14.  65460  min.          17.  1153800  sec. 
18.   Reduce  f  of  a  minute  to  the  fraction  of  a  day. 


MEASURES  OF  VALUE.  187 

CIRCULAR  OR  ANGULAR  MEASURE. 
EXERCISES. 

229.  1.    How  many  minutes  are  there  in  5°  ?     In  7°  ? 

2.  How  many  seconds  are  there  in  3'  ?     In  5'  ?     In  6'  ? 

3.  How  many  degrees  are  there  in  \  cir.  ?     In  J  cir.  ? 

4.  How  many  seconds  are  there  in  34°  12'  43"  ? 

5.  Reduce  468560  sec.  to  higher  denominations. 

6.  Eeduce  195600  sec.  to  signs. 

7.  Eeduce  35°  41'  18"  to  seconds ;  also  18°  37'  14". 

Eeduce  to  higher  denominations : 

8.  489600".  10.    248300".  12.   486300". 

9.  381400".  11.   319400".  13.   389600". 

MEASURES  OF  VALUE. 
ENGLISH  MONEY. 

230.  1.   How  many  farthings  are  there  in  a  penny?     In 
8  pence  ?     In  10  pence  ?     In  15  pence  ?     In  20  pence  ? 

2.  How  many  pence   are   there   in  2   shillings  ?    In  5 
shillings?     In  6s.  3d.?     In  5s.  10  d?     In  8s.  4 d.? 

3.  How  many  shillings   are  there  in  3  pounds?    In  5 
pounds?   In  £3  5s.  ?   In  £4  8s.?   In  £5  10s.? 

4.  How  many  pence  are  there  in  £  1  ?   In  £  ^  ?  In  £  ^  ? 

5.  How  many  pence  are  there  in  40  far.  ?     In  48  far.  ? 

6.  How  many  shillings  are  there  in  48  d.  ?    In  60  d.  ? 

WRITTEN  EXERCISES. 

Eeduce  to  units  of  lower  denominations  : 

1.  £2    10s.    6d  6.  £45  3s.  9fd. 

2.  £13    5s.  5fd.  7.  £75  to  farthings. 

3.  £14    6s.  5Jd.  8.  |  of  a  sovereign. 

4.  £  20  12s.  6f  d.  9.  .65  of  a  pound. 

5.  £36    6s.  8£cZ.  10.  f  of  a  guinea. 


188  DENOMINATE  NUMBERS. 

Reduce  to  units  of  higher  denominations : 

11.  34567  far.  14.  35968  far.  17.  48596  far. 

12.  21586 d.  15.  16500 d.  18.  34856d 

13.  3846  far.  16.  47384 d  19.  12000 d. 

20.  Reduce  3s.  6d  to  the  decimal  of  a  pound. 

21.  Reduce  5s.  Sd.  to  the  decimal  of  a  pound. 

22.  Express  £  3  6  s.  5  d.  as  pounds  and  decimals  of  a  pound. 

23.  Express  £  5  8  s.  4  d  as  pounds  and  decimals  of  a  pound. 
Find  the  value  of  the  following  in  U.  S.  money : 

24.  £25  18s.  6d 

SOLUTION.  —     £  25  18  s.  6  d.  -  £  25.925. 
£1  =  $4.8665. 
.-.  £25.925  =  $4.8665  x  25.925  =  126.164. 

25.  £31     6s.  5d.      28.  £51  6s.  31.  £10  6s.  5d 

26.  £24     8s.  3d.      29.  £35  8s.  Sd.      32.  £35  8s.  2d. 

27.  £29  15s.  6d      30.  £18  9s.  4d      33.  £15  7s.  4d 
Find  the  value  of  the  following  in  English  money : 

34.  $  395.18.     36.  $573.25.     38.  $237.16.     40.  $324.25. 

35.  $246.93.     37.  $615.86.     39.  $426.95.     41.  $1000. 

MISCELLANEOUS  TABLES. 

231.   1.   How  many  are  5  dozen  ?     8  dozen  ?     10  dozen  ? 

2.  How  many  are  4  score  ?     3  score  ?    3  score  and  10  ? 

3.  How  many  is  a  gross  ?     How  many  are  2  gross?     3 
gross  ?     How  many  is  a  great  gross  ? 

4.  How  many  sheets  are  there  in  3  quires  ?   In  4  quires  ? 

5.  How  many  quires  are  there  in  3  reams  ?    In  5  reams  ? 

6.  How  many  reams  are  there  in  3  bundles  ? 

7.  How  many  knives  are  there  in  6  sets  ?     In  8  sets  ? 

8.  How  many  dozen  are  there  in  60  things  ?     In  72  ? 


ADDITION.  189 


ADDITION. 

232.  The  processes  of  adding,  subtracting,  multiplying, 
and  dividing  compound  denominate  numbers  are  based  upon 
the  same  principles  as  those  governing  similar  operations 
in  simple  numbers. 

The  only  difference  between  the  processes  is  caused  by 
compound  numbers  having  a  varying  scale,  while  simple 
numbers  have  a  uniform  decimal  scale. 

WRITTEN    EXERCISES. 

1.  What  is  the  sum  of  130  rd.  5  yd.  1  ft.  6  in.,  215  rd. 
2  ft.  8  in.,  304  rd.  4  yd.  11  in.  ? 

v  rd.       yd.      ft.     in.          EXPLANATION.  —  The    numbers 

130       516    'should  be  written  as  in  simple  ad- 

O-JK        Q        o        8     dition>  so  tnat  units  of  tne  same 

denomination  stand  in  the  same 

304       4       0     11     column,  and  for  convenience  we 


9       i          10         Jl       9         1       kegin  at  the  ri&ht  t0  add' 

±         The  sum  of  the  inches  is  25  in., 

•J=l       6     which  is  equal  to  2  ft.  1  in.    We 

7 !        77        3       7 7     write  the  1  under  the  inches  and 

add  the  2  ft.  to  the  feet.    The  sum 

of  the  feet  is  5  ft.,  or  1  yd.  2  ft.     We  write  the  2  as  feet  La  the  sum 
and  add  the  1  yd.  to  the  yards. 

The  sum  of  the  yards  is  10  yd.,  or  1  rd.  4|yd.  We  write  the  4£  yd. 
as  yards  in  the  sum,  and  add  the  1  rd.  to  the  rods.  The  sum  of  the 
rods  is  650  rd.,  or  2  mi.  10  rd.,  which  we  write  as  miles  and  rods  in  the 
sum. 

Therefore  the  sum  is  2  mi.  10  rd.  4J  yd.  2  ft.  1  in.  Or,  since  \  yd. 
equals  1  ft.  6  in.,  the  sum  may  be  expressed  as  2  mi.  10  rd.  5  yd.  7  in. 

2.  Find  the  sum  of  18  Ib.  10  oz.  14  pwt.  20  gr.,  28  Ib. 
6  oz.  15  pwt.  15  gr.,  36  Ib.  4  oz.  12  pwt.  16  gr. 

3.  Find  the  sum  of  12  bu.  3  pk.  7  qt.,  25  bu.  5  qt.,  8  bu. 
2  pk.  1  pt.,  48  bu.  3  pk.,  42  bu.  1  pk.  2  qt.,  48  bu.  3  pk.  6  qt. 

4.  Find  the  sum  of  18  T.  12  cwt.  50  Ib.  15  oz.,  25  T.  12  cwt. 
19  Ib.  13  oz.,  15  T.  14  cwt.  35  Ib.  9  oz.,  20  T.  18  cwt. 


190  DENOMINATE   NUMBERS. 

5.  Find  the  sum  of  14  gal.  3  qt.  1  pt.  3  gi.,  15  gal.  2  qt. 
1  pt.  2  gi.,  11  gal.  2  qt.  2  gi.,  16  gal.  1  pt.,  30  gal.  3  qt.  2  gi. 

6.  What  is  the  sum  of  28  tb  735323  15  gr.,  251b  105 
43  13  15  gr.,  191b  95   53  13  23  gr.,  27lb  83   33  23 
17  gr.,  24lb73  23  13  18  gr.  ? 

7.  Find  the  sum  of  9  mi.  212  rd.  2  yd.  2  ft.  8  in.,  10  mi. 
185  rd.  3  yd.  9  in.,  15  mi.  76  rd.  1  yd.  3  ft.  5  in.,  20  mi. 
200  rd.  4  yd.  6  in.,  36  mi.  126  rd.  5  yd.  2  ft.  10  in. 

8.  Find  the  sum  of  132  sq.  rd.  20  sq.  yd.  8  sq.  ft.  72  sq. 
in.,  12  sq.  rd.  15  sq.  yd.  7  sq.  ft.  80  sq.  in.,  18  sq.  yd.  6  sq. 
ft.  86  sq.  in. 

9.  What  is  the  sum  of  £45  5s.  3fd,  £36   8s.  5^cZ., 
£65  15s.  7H.,  £52  13s.  9f  d,  £120  10s.  Sd.  ? 

10.  Find  the  sum  of  142  cu,  yd.  18  cu.  ft.  1229  cu.  in., 
275  cu.  yd.  25  cu.  ft.  1076  cu.  in.,  382  cu.  yd.  17  cu.  ft.  1521 
cu.  in.,  420  cu.  yd.  20  cu.  ft.  1507  cu.  in. 

11.  New  York  is  74°  3'  west  longitude,  and  Paris,  France, 
is  2°  20'  east.     What  is  the  difference  in  longitude  between 
the  two  cities  ? 

SUGGESTION.  —  To  find  the  difference  in  longitude  between  two 
places,  one  of  which  is  east  and  the  other  west  longitude,  we  add 
their  respective  longitudes. 

12.  Find  the  sum  of  f  mi.,  .35  rd.,  and  2|  rd. 

SOLUTION. 

rd.         ft.  in. 

f  mi.  =  137        2  4f 

.35  rd.  =  5  9r% 

2frd.  =     2        6  2f 


139      14 

13.  Find  the  sum  of  10  wk.  4  da.  5J  hr.,  2  wk.  5f  da., 
9  wk.  3  da.  18  hr.  12|  min.,  6  da.  15  hr.  15  min. 

14.  What  is  the  sum  of  £.35,  £2.875,  and  £6  8s.  6d? 

15.  What  is  the  sum  of  f  T.,  £  cwt.,  and  f  Ib.  ? 


SUBTRACTION.  191 

SUBTRACTION. 
WRITTEN    EXERCISES. 

233.   1.   From  127  rd.  3  yd.  1  ft.  7  in.,  subtract  100  rd. 
4  yd.  2  ft.  9  in. 

EXPLANATION.  —  The  numbers  should 

rd.  yd.  ft.  in.  be  written  as  in  simple  subtraction,  so  that 
127  317  units  of  the  same  order  stand  in  the  same 
100  429  column,  and,  for  convenience,  we  should 

begin  at  the  right  to  subtract. 

26       3£     1     10          Since  9  in.  cannot  be  subtracted  from 
=1       6      7  in.,  we  add  to  7  in.  a  unit  of  the  next 
higher  order,  making  1  ft.  7  in.,  or  19  in. 


26       4       0       4     Then  9  in.  taken  from  19  in.  leave  10  in., 
which  we  write  as  inches  in  the  remain- 
der.   Inasmuch  as  1  ft.  was  added  to  7  in.,  there  are  no  feet  remaining 
in  the  minuend. 

Since  we  cannot  subtract  2  ft.  from  0  ft.,  we  add  to  0  ft.  a  unit  of 
the  next  higher  order,  making  3  ft.  Then  2  ft.  taken  from  3  ft.  leave 

1  ft.,  which  we  write  as  the  feet  of  the  remainder. 

Since  4  yd.  cannot  be  subtracted  from  2  yd.,  we  add  to  2  yd.  a 
unit  of  the  next  higher  order  and  proceed  as  before.  The  remainder 
is  26  rd.  4  yd.  0  ft.  4  in. 

2.  From  16  Ib.  10  oz.  16  pwt.  18  gr.,  take  12  Ib.  11  oz. 
17  pwt.  15  gr. 

3.  From  £25  4s.,  take  £20  8s.  lOd 

4.  From  5  Ib  7  5,  take  3  Ib  10  I  5  3  1  3  15  gr. 

5.  From  2  hhd.  20  gal.  3  qt.,  take  1  hhd.  60  gal.  3  qt.  1  pt. 

6.  From  4  mi.  126  rd.  4  yd.  6  in.,  take  2  mi.  140  rd.  3  yd. 

2  ft.  8  in. 

7.  From  a  barrel  containing  36  gal.  3  qt.  1  pt.  of  vine- 
gar, there  were  sold  27  gal.  1  qt.  1  pt.  2  gi.      How  much, 
remained  unsold  ? 

8.  One  train  left  Albany  at  7  o'clock  35  min.  A.M.,  and 
another  at  11  o'clock  15  min.  A.M.     How  long  after  the  first 
did  the  second  start  ? 


192  DENOMINATE  NUMBERS. 

9.  From  f  bbl.  take  7|  gaL 

SOLUTION. 
gal.     qt.     pt.     gi. 
|  bbl.  =  23       2       1 
7|  gal.  =    7      2      0      8£ 

16      0      0        f 

10.  From  £|£,  take  5s.  2d.  3  far. 

11.  From  |  mi.,  take  120.65  rd. 

12.  From  -J  wk.,  take  T4^-  da. 

13.  From  15.576  bu.,  take  3.65  pk. 

14.  From  .9  mi.,  take  120  rd.  4  yd.  2  ft. 

15.  Baltimore  is  76°  37'  west  longitude,  and  San  Francisco 
is  122°26Jf  west  longitude.     What  is  their  difference  in 
longitude  ? 

SUGGESTION. — To  find  the  difference  in  longitude  between  two 
places  both  of  which  are  east,  or  both  west,  we  subtract  the  less  from 
the  greater. 

16.  How  long  was  it  from  Jan.  10, 1841,  to  May  7, 1853  ? 

yr.       mo.      da.  EXPLANATION. — Since  the  later  date  expresses 

1853      5         7      the  greater  period  of  time,  we  write  it  as  the  minu- 
1841       1       10      en(*»  an(* tne  earlier  date  as  the  subtrahend,  giving 

— ^2 o ^      the  month  its  number  instead  of  the  name.     We 

then  subtract  as  in  denominate  numbers,  con- 
sidering 30  days  1  month,  and  12  months  one  year.  The  remainder 
will  be  the  time  as  correct  as  it  can  be  expressed  in  months  and  days. 

17.  A  certain  person  was  born  June  24,  1859.     How  old 
was  he  Sept.  9,  1891? 

18.  How  many  years,  months,  and  days  is  it  from  the 
day  of  your  birth,  or,  how  old  are  you  ? 

19.  A  note  dated  May  6,  1885,  was  paid  Nov.  4,  1890. 
How  long  did  it  run  before  it  was  paid  ? 

20.  A  man  was  born  Feb.  29,  1844,  and  died  Mar.  15, 
1880.     How  many  birthdays  did  he  have,  and  what  was  his 
age? 


MULTIPLICATION.  193 

MULTIPLICATION. 
WRITTEN    EXERCISES. 

234.    1.    How  much  is  5  times  147  rd.  4  yd.  2  ft.  8  in.  ? 

EXPLANATION.  —  We  write  the  numbers  as 
VAT  yl    9    »       in  simPle  numbers,  and  for  convenience  begin 

at  the  right  to  multiply. 
5  5  times  8  in.  are  40  in.,  or  3  ft.  4  in.    We 


2  mi.  99  2  1  4  write  the  4  in.  as  inches  in  the  product,  and 
reserve  the  3  ft.  to  add  to  the  product  of  feet. 

5  times  2  ft.  are  10  ft. ;  10  ft.  +  3  ft.  reserved  equal  13  ft.,  or  4  yd. 
1  ft.  We  write  the  1  ft.  in  the  product,  and  reserve  the  4  yd.  to  add  to 
the  product  of  yards. 

5  times  4  yd.  equal  20  yd.  ;  20  yd.  +  4  yd.  reserved  equal  24  yd.,  or 
4  rd.  2  yd.  We  write  the  2  yd.  in  the  product,  and  reserve  the  rods  to 
add  to  the  product  of  rods. 

5  times  147  rd.  are  735  rd.  ;  735  rd.  4-  4  rd.  reserved  equal  739  rd., 
or  2  mi.  99  rd.,  which  we  write  in  the  product. 

Therefore  the  product  is  2  mi.  99  rd.  2  yd.  1  ft.  4  in. 

2.  Multiply  12  bu.  3  pk.  2  qt.  1  pt.  by  8. 

3.  Multiply  7  Ib.  8  oz.  15  pwt.  18  gr.  by  15. 

4.  Multiply  £4  8s.  6  d.  by  5. 

5.  Multiply  3  cwt.  10  Ib.  9  oz.  by  12. 

6.  Multiply  38  gal.  3  qt.  1  pt.  2  gi.  by  10. 

7.  How  much  wheat  can  be  put  into  18  sacks,  if  each 
sack  holds  1  bu.  3  pk.  7  qt.  1  pt.  ? 

8.  A  solar  year  consists  of  365  da.  5  hr.  48  min.  49.7 
sec.     How  much  time  is  there  in  20  solar  years  ? 

9.  If  one  silver  spoon  weighs  3  oz.  10  pwt.  15  gr.,  what 
will  be  the  weight  of  two  sets  of  6  spoons  each  ? 

10.  How  many  bushels  of  corn  will  a  field  of  14  acres 
produce,  if  it  produces  28  bu.  3  pk.  5  qt.  1  pt.  to  the  acre  ? 

11.  If  one  load  of  wood  measures  112  cu.  ft.  432  cu.  in., 
how  much  will  25  loads  of  the  same  size  measure  ? 

STAND.   AR. 13 


194  DENOMINATE  NUMBERS. 

DIVISION. 
WRITTEN  EXERCISES. 
235.   1.   Divide  27  bu.  3  pk.  5  qt.  1  pt.  into  6  equal  parts. 

/j\rv7-u      0     -,     K     ,1  EXPLANATION.  —  Since  the  quantity 

6)27  bu.  3  pk.  5  gt.  1  pt.     ig to  be  diyided  into  6  equal  pa^  eac£ 

4  2  4  1|-  part  will  contain  one  sixth  of  the  quan- 
tity. 

One  sixth  of  27  bu.  is  4  bu.,  with  a  remainder  of  3  bu.  We  write 
the  4  bu.  in  the  quotient  and  add  the  3  bu.  remaining  to  the  number 
of  the  next  lower  denomination,  making  15  pk. 

One  sixth  of  15  pk.  is  2  pk.,  with  3  pk.  remaining.  We  write  the 
2  pk.  in  the  quotient,  and  add  the  3  pk.  remaining  to  the  number  of 
the  next  lower  denomination,  making  29  qt. 

One  sixth  of  29  qt.  is  4  qt.,  with  5  qt.  remaining.  We  write  the  4 
qt.  in  the  quotient,  and  add  the  5  qt.  to  the  number  of  the  next  lower 
denomination,  making  11  pt. 

One  sixth  of  11  pt.  is  If  pt.,  which  we  write  in  the  quotient. 

Therefore  the  quotient  is  4  bu.  2  pk.  4  qt.  If  pt. 

2.  Divide  70  Ib.  10  oz.  14  pwt.  12  gr.  by  6. 

3.  Divide  112  T.  16  cwt.  66  Ib.  by  7. 

4.  Divide  117  hhd.  33  gal.  2  qt.  1  pt.  2  gi.  by  9. 

5.  Divide  153  mi.  313  rd.  3  yd.  2  ft.  by  11. 

6.  Divide  103  C.  12  cu.  ft.  632  cu.  in.  by  10. 

7.  If  15  bars  of  silver  weigh  39  Ib.  8  oz.  16  pwt.,  what 
is  the  average  weight  of  a  bar  ? 

8.  If  6  men  build  124  rd.  2  ft.  6  in.  of  wall  in  17  days, 
how  much  do  they  build  in  one  day  ? 

9.  A  man  traveled  348  mi.  52  rd.  in  28  days.     How  far, 
on  an  average,  did  he  travel  per  day  ? 

.     10.   The  entire  weight  of  41  hhd.  of  sugar  is  19  T.  6  cwt. 
22  Ib.     What  is  the  average  weight  of  a  hhd.  ? 
,     11.   If  a  ship  sailed  64°  59'  3"  in  29  days,  how  far  did  she 
sail  on  an  average  per  day  ? 


DIVISION.  195 

12.  If  31  cwt.  18  Ib.  of  tea  are  put  up  in  packages,  each 
containing  3  Ib.  8  oz.,  how  many  packages  will  there  be  ? 

SOLUTION. 

31  cwt.  18  Ib.  =  49888  oz. 

3  Ib.  8  oz.  =  56  oz. 
49888  oz.  -5-  56  oz.  =  890f ,  the  number  of  packages. 

13.  Divide  £36  13s.  3d.  by  £5  4s.  9d. 

14.  Divide  2  Ib.  7  oz.  19  pwt.  by  5  oz.  6  pwt.  12  gr. 

15.  Divide  40  T.  16  cwt.  11  Ib.  4  oz.  by  2  T.  14  cwt. 
10  Ib.  12  oz. 

16.  How  long  will  9  bu.  1  pk.  4  qt.  of  oats  last  a  horse 
if  he  eats  1  pk.  4  qt.  per  day  ? 

17.  How  many  sacks  will   it  require  to  contain  39  bu. 
2  pk.  6  qt.  of  wheat,  if  each  sack  holds  1  bu.  3  pk.  1\  qt.  ? 

18.  If  one  bale  of  hay  weighs  5  cwt.  25  Ib.,  how  many 
bales  will  it  take  to  weigh  2  T.  7  cwt.  25  Ib.  ? 

19.  A  man  traveled  3  mi.  20  rd.  4  yd.  in  one  hour.     In 
what  time  could  he  travel  240  miles,  traveling  at  the  same 
rate? 

20.  A  man  bought  60  cwt.  85  Ib.  of  sugar  at  4  cts.  a 
pound.     He  sold  \  of  it  at  5  cts.  a  pound,  \  of  it  at  5£  cts. 
a  pound,  and  the  remainder  at  cost.    How  much  did  he  gain  ? 

21.  How  many  fence  pickets,  2  ft.  4  in.  long  and  2  in. 
wide,  can  be  made  from  8  boards,  each  11  ft.  8  in.  long  and 
8  in.  wide  ? 

22.  How  many  medals,  each  weighing  5  oz.  13  pwt.  21  gr., 
can  be  made  from  a  bar  of  gold,  which  weighs  88  Ib.  8  oz. 
14  pwt.  15  gr.  ? 

23.  A  man  owned  a  pile  of  wood  containing  40  cords.     If 
it  was  4  ft.  wide  and  8  ft.  high,  what  was  its  length  ? 


196  DENOMINATE   NUMBERS. 

REVIEW   EXERCISES. 

236.   1.   What  will  be  the  cost  of  8  Ib.  6  oz.  of  lard  at  10 
cents  per  pound  ? 

2.  How  much  will  2  pk.  3  qt.  of  beans  cost  at  12  cents 
a  quart  ? 

3.  How  many  quart  boxes  will  2  bu.  3  pk.  5  qt.  of  straw- 
berries fill  ? 

4.  A  dealer  sold  1  bu.  5  qt.  of  chestnuts  at  5  cents  a 
pint.     How  much  did  he  get  for  them  ? 

5.  How  many  bushels  of  potatoes,  at  $  .45  a  bushel,  must 
be  given  for  5  gal.  3  qt.  of  syrup  at  15  cents  a  quart  ? 

6.  How  many  square  yards  are  there  in  a  lot  75  feet 
long  and  60  feet  wide  ? 

7.  The  area  of  a  blackboard  3f  feet  wide  is  1121  Sq.  ft. 
What  is  its  length  ? 

8.  The  diameter  of  the  earth  is  7912  miles.     How  many 
feet  is  it  ? 

9.  How  high  is  a  horse  that  measures  15  hands  ? 

10.  A  farm  is  67  ch.  83  1.  long.     How  many  rods  long 
is  it? 

11.  How  many  inches  are  there  in  59  ch.  75  1.  ? 

12.  A  pasture  containing  10  acres  had  a  width  of  20  rods. 
How  long  was  it  ? 

13.  What  is  the  difference  between  10  square  feet  and 
10  feet  square  ?    Illustrate  this  by  drawings. 

14.  What  will  be  the  expense  of  painting  a  roof  48  feet 
long  and  22  feet  wide  at  $  .30  a  square  yard  ? 

15.  How  much  will  8  barrels  of  flour  cost,  at  the  rate  of 
&J-  cents  a  pound  ? 


REVIEW  EXERCISES.  197 

16.  How  many  bars  of  iron,  each  weighing  41  Ib.  10 J  oz., 
will  it  take  to  weigh  a  ton  ? 

17.  How  many  times  will  a  wheel  12  ft.  4  in.  in  circum- 
ference revolve  in  going  10  miles  ? 

18.  Milton  was  born  Dec.  9,  1608,  and  died  Nov.  8, 1675. 
What  was  his  age  at  the  time  of  his  death  ? 

19.  Prom  a  pile  of  wood  containing  120  cords,  there  were 
sold  at  one  time  48  C.  96  cu.  ft.,  and  at  another  time  36  C. 

28  cu.  ft.     How  much  remained  ? 

20.  A  man  who  owned  300  acres  of  land,  sold  120  A. 

29  sq.  rd.  27  sq.  yd.  6  sq.  ft.     How  much  remained  unsold  ? 

21.  Reduce  ^  of  a  bushel  to  lower  denominations. 

22.  Reduce  -f-  of  a  scruple  to  the  fraction  of  a  pound. 

23.  Reduce  ^-  of  an  acre  to  lower  denominations. 

24.  What  is  the  difference  between  6  dozen  dozen  and  a 
half  a  dozen  dozen  ? 

25.  Express  .65  of  a  pint  as  a  decimal  of  a  bushel. 

26.  How  many  times  must  a  man  dip  with  a  dipper  hold- 
ing 1  qt.  1  pt.  so  that  he  may  empty  a  cask  containing  31-J- 
gaL? 

27.  How  many  barrels  of  sugar,  each  containing  2  cwt. 
35  Ib.,  are  there  in  3  T.  4  cwt.  18  Ib.  ? 

28.  Express  .375  of  a  week  as  a  fraction  of  a  year. 

29.  What  part  of  5  gal.  3  qt.  1  pt.  are  2  gal.  1  qt.  1  pt.  ? 

30.  Reduce  4  hr.  15  min.  to  the  decimal  of  a  day. 

31.  Reduce  .1845  of  a  gill  to  the  decimal  of  a  gal. 

32.  What  part  of  9  inches  square  are  9  square  inches  ? 

33.  From  .625  Ib.  Troy  take  5.25  oz.  Troy. 

34.  What  is  the  length  of  a  fence  inclosing  a  square  field, 
each  side  of  which  is  25  rd.  3  yd.  2-f  ft.  long  ? 


198  DENOMINATE   NUMBERS. 

35.  If  a  hogshead  of  sugar  weighs  5  cwt.  24  Ib.  4  oz., 
what  will  8  hhd.  be  worth  at  4£  ct.  per  pound  ? 

86.  How  many  cups,  holding  one  half  pint  each,  can  be 
filled  from  a  coffee  urn  holding  2  gal.  3  qt.  1^  pt.  ? 

37.  If  the  weight  of  a  bushel  of  wheat  is  60  Ib.,  how 
many  bags  that  hold  2  bu.  each  will  be  required  to  sack 
3  T.  4  cwt.  20  Ib.  of  wheat  ? 

38.  Which  is  the  heavier,  and  how  much,  a  pound  of 
lead  or  a  pound  of  gold  ? 

39.  Which  is  the  heavier,  and  how  much,  an  ounce  of 
feathers  or  an  ounce  of  silver  ? 

40.  An   iron   block  weighed    115    pounds   Avoirdupois 
weight.     How  much  would  it  have  weighed  by  Troy  weight  ? 

41.  How  many  steel  rails  30  ft.  long  are  needed  to  build 
5  miles  of  railroad  ? 

42.  If  a  horse  travels  on  an  average  a  mile  in  10  min. 
15  sec.,  how  far  does  he  go  in  6  hr.  ? 

43.  I  have  a  rectangular  farm  230  rods  long  and  180  rods 
wide  which  is  worth  $  75  per  acre.     What  is  the  value  of 
the  farm  ? 

44.  Find  the  sum  of  £  mi.,  £  rd.,  |  ft. 

45.  What  part  of  a  mile  is  f  of  6  rd.  3  yd.  2  in.  ? 

46.  Find  the  value  in  U.  S.  money  of  the  contents  of  a 
purse    containing    35    sovereigns,    27    half-sovereigns,   13 
crowns,  41  half-crowns,  a  guinea,  and  a  shilling. 

47.  A  farmer  sowed  4  bu.  1  pk.  1  qt.  of  seed,  and  har- 
vested from  it  110  bu.  3  pk.  5  qt.     How  much  did  he  raise 
from  a  bushel  of  seed  ? 

48.  What  will  be  the  cost  of  3  T.  6  cwt.  27  Ib.  of  coal 
at  $4.75  a  ton? 

49.  A  merchant's  profits  in  1\  months  were  $1675.     At 
this  rate,  what  would  be  his  profits  in  a  year  ? 


REVIEW  EXEKCISES.  199 

50.  A  train  running  from  Philadelphia  to  New  York,  a 
distance  of  90  miles,  makes  the  whole  distance  in  1  hr. 
35  min.     What  is  its  rate  per  hour  ? 

51.  If  a  cubic  foot  of  ice  weighs  57f  pounds,  how  many 
cubic  feet  of  ice  will  it  take  to  weigh  a  ton  ? 

52.  Washington  is  77°  2' 48"  west  longitude,  and  Albany 
is  73°  44' 53"  west  longitude.      What  is  the   difference  in 
longitude  between  these  two  places  ? 

53.  Paris  is  2°  20' 22"  east  of  Greenwich,  and  New  York 
is  74°  3"  west.     What  is  their  difference  in  longitude  ? 

54.  What  will  be  the  cost  of  fencing  a  rectangular  lot 
18  rods  by  24  rods  at  18|  cents  a  foot  ? 

55.  How  much  fertilizer  will  be  needed  for  5  A.  96  sq. 
rd.  of  land,  allowing  3  bu.  1  pk.  3  qt.  to  an  acre  ? 

56.  The  area  of  a  rectangular  field  is  60  A.  130  sq.  rd., 
and  one  side  is  20.25  chains.     What  is  the  length  of  the 
other  side  ? 

57.  How  many  days,  of  10  hours  each,  will  it  take  to 
count  a  million  at  the  rate  of  100  a  minute  ? 

58.  What  will  be  the  cost  of  5  reams  15  quires  and  20 
sheets  of  paper  at  $3.60  a  ream  ? 

59.  How  many  barrels  of  flour,  at  $  4.75  per  barrel,  must 
be  given  for  3  T.  5  cwt.  of  coal  at  $  6.50  per  ton  ? 

60.  A  druggist  purchased  9J  ounces   of  quinine  at  $  .40 
an  ounce  Avoirdupois,  and  sold  it  at  $  .60  an  ounce  Troy. 
How  much  did  he  gain  ? 

61.  How  many  silver  spoons,  each  weighing  2  oz.  5  pwt.? 
can  be  made  from  a  bar  of  silver  weighing  6  Ib.  4  oz.  10  pwt.  ? 

62.  I  wish  to  have  6  reams  15  quires  20  sheets  of  paper 
printed  for  one  fourth  sheet  posters.     How  many  can  I  get, 
and  what  will  they  cost  at  $  5.75  per  M  ? 


200  DENOMINATE  NUMBERS. 

63.  If  I  burn  a  pint  of  kerosene  every  night,  what  will  a 
three  weeks'  supply  cost  me  at  15  cents  a  gallon  ? 

64.  What  is  the  value  in  U.  S.  money  of  1000  francs  ? 

65.  A  man  retails  oil  at  10  cts.  a  pint.     What  is  his 
profit  on  3  bbl.,  of  31^  gal.  each,  which  cost  $  .65  a  gallon  ? 

66.  What  will  12  horses  cost  in  U.  S.  money,  if  5  horses 
cost  £175  10s.  6d.  ? 

67.  A  milkman  sold  one   morning  220  qt.  of  milk  at 
6  cts.  a  quart.     His  measure  lacked  %  of  a  gill  of  holding  a 
full  quart.     What  was  the  actual  worth  of  the  milk  sold  ? 

68.  How  many  powders,  of  6  grains  each,  can  be  made 
from  one  fourth  of  an  ounce  of  medicine  ? 

69.  A  physician's  prescription  calls  for  3vij,  3ij  of  calo- 
mel.    How  many  pills,  of  5  grains  each,  can  be  made  from 
the  prescribed  quantity  ? 

70.  If  a  grocer's  scales  give  only  15|  oz.  for  a  pound,  of 
how  much  money  does  he  defraud  his  customers  in  the  sale 
of  5  bbl.  of  sugar,  each  weighing  2  cwt.  10  Ib.  12  oz.,  true 
weight,  at  5  cents  a  pound  ? 

71.  A  man  sold  8  bu.  3  pk.  4  qt.  of  cranberries  at  $3|-  a 
bushel,  and  took  his  pay  in  flour  at  3|-  cents  a  pound.    How 
many  barrels  of  flour  did  he  receive  ? 

72.  A  man  purchased  54  cwt.  85  Ib.  of  sugar  at  4  cts.  a 
pound.     He  sold  £  of  it  at  5  cts.  a  pound,  £  of  it  at  5£  cts. 
a  pound,  and  the  rest  at  cost.     How  much  did  he  gain  ? 

73.  How  many  silver  coins,  each  weighing  412£  gr.,  can  be 
coined  from  a  bar  of  silver  weighing  8  Ib.  4  oz.  Avoirdupois  ? 

74.  I  wish  to  put  116  bu.  1  pk.  4  qt.  of  grain  into  bags 
that  shall  contain  2  bu.  1  pk.  4  qt.  each.     How  many  bags 
will  be  required  ? 


LONGITUDE   AND   TIME. 


237.   1.   Where  does  the  suii  appear  to  rise  ? 

2.  How  often  does  it  appear  to  rise  in  the  east  ? 

3.  Through  how  many  degrees  of  space  does  it  appear 
to  pass  in  this  daily  motion  ?  Ans.  360°. 

4.  Since  it  seems  to  travel  360°  in  one  day,  or  24  hours, 
how  great  will  be  its  apparent  motion  in  1  hour  ? 

5.  If  the  earth  moves  15°  in  one  hour,  how  far  will  it 
move  in  1  minute  ? 

6.  If  it  moves  £  of  a  degree  or  15'  of  distance  in  one 
minute  of  time,  how  far  will  it  move  in  1  second  of  time  ? 

7.  How  does  the  number  of  degrees  passed  over  compare 
with  the  number  of  hours  ?    The  number  of  minutes  of  space 
with,  the  number  of  minutes  of  time  ?     The  number  of  sec- 
onds of  space  with  the  number  of  seconds  of  time  ? 

8.  When  it  is  sunrise  at  any  place,  how  long  will  it  be 
before   it  is  sunrise  at  a  place   15°  west  of  that  place? 
30°  west  ?     45°  west  ? 

9.  When  it  is  sunrise  at  any  place,  how  long  before  was 
it  sunrise  at  a  place  15°  east  of  that  place  ?    30°  east  ? 

10.  When  it  is  noon  at  any  place,  what  time  is  it  at  a 
place  15°  west  ?    15°  east  ?    30°  west  ?    30°  east  ? 

11.  If  I  travel  eastward,  will  my  watch  become  too  slow 
or  too  fast?     If  I  travel  westward,  will  my  watch  be  too 
slow  or  too  fast  ? 

201 


202  LONGITUDE   AND   TIME. 

12.    What  places  have  noon  at  the  same  time?    Midnight? 

238.  A  Meridian  is  an  imaginary  line  passing  from  the 
North  Pole  to  the  South  Pole,  through  any  place. 

239.  The  distance  east  or  west  from  a  given  meridian  is 
called  Longitude. 

The  meridians  from  which  longitude  is  commonly  reckoned  are  those 
which  pass  through  Washington,  D.C.,  and  Greenwich,  England. 

RELATIONS  BET-WEEN  LONGITUDE  AND 
TIME. 

Two  places  distant  from  each  other 

15°  of  longitude  differ  1  hour       in  time. 

15'  "          «  "      1  minute    "      " 

15"  «          "  "      1  second     "      " 

1°  "          "  «      4  minutes  "      " 

V  "          "  "      4  seconds   "      " 

WRITTEN    EXERCISES. 

240.  To  find  the  difference  in  time  between  two  places  when 
the  difference  in  longitude  is  given. 

1.  Two  places  are  35°  12'  15"  apart.     What  is  the  dif- 
ference in  time  between  them  ? 

15)35°  12*  15"         EXPLANATION.  —  Since  places  distant  from  each 

2  20  49  °ther  15°  °f  lon£itude  differ  *  nr-  in  time»  15/  of 
longitude  1  min.  in  time,  and  15"  of  longitude  1  sec. 
in  time,  ^  of  the  numbers  representing  difference  in  degrees,  minutes, 
and  seconds  of  longitude  will  equal  the  numbers  representing  the  dif- 
ference in  hours,  minutes,  and  seconds  of  tune.  Therefore,  the  differ- 
ence in  time  is  2  hr.  20  min.  49  sec. 

2.  The  difference  in  longitude  between  two  places  is 
46°  15'  30".     What  is  the  difference  in  time  ? 

3.  Washington  is  77°  2'  48"  west  from  Greenwich.    What 
is  the  difference  in  time  between  the  two  places  ? 


WRITTEN   EXERCISES.  203 

4.  New  York  is  74°  3"  west  longitude,  and  Chicago  is 
87°  38'  west.     What  is  their  difference  in  time  ? 

5.  The  longitude  of  Philadelphia  is  75°  10'  west  from 
Greenwich,  and  that  of  San  Francisco  122°  26'  15"  west  from 
Greenwich.     What  time  is  it  at  San  Francisco  when  it  is 
noon  at  Philadelphia  ? 

6.  Boston  is  5°  59' 18"  east  from  Washington.    What 
time  is  it  at  Washington  when  it  is  noon  at  Boston  ? 

7.  The   longitude   of   Albany  is  73°  44' 53"  west  from 
Greenwich,  and  that  of  St.  Paul  is  93°  4' 55"  west  from 
Greenwich.    How  much  earlier  does  the  sun  rise  at  Albany 
than  at  St.  Paul  ? 

8.  The    longitude   of   Berlin  is   13°  23' 43"   east   from 
Greenwich,  and  that  of  Cincinnati  84°  26'  west  from  Green- 
wich.    What  is  their  difference  in  time  ? 

9.  Paris   is   2°  20' 22"   east   from   Greenwich.     Will   a 
traveler's  watch  be  slow  or  fast,  and  how  much,  when  he 
goes  from  Greenwich  to  Paris  ? 

10.  Pekin  in  China  is  116°  27' 30"  east  longitude,  and 
Washington  is  77°  west  longitude.  When  it  is  midnight 
Dec,  31  at  Washington,  what  time  is  it  at  Pekin  ? 

241.  To  find  the  difference  in  longitude  between  two  places 
when  the  difference  in  time  is  given. 

1.    The  difference  in  time  between  two  places  is  2  hr. 
20  min.  49  sec.     What  is  their  difference  in  longitude  ? 

2  hr.  20  min.  49  sec.         EXPLANATION.  —  Since  &ere  are  15  times 
JK  as  many  degrees,  minutes,  and  seconds  of 

— — longitude,  as  there  are  hours,  minutes,  and 

35      1^  15  seconds  of  time,  15  times  the  numbers  rep- 

resenting the  difference  in  hours,  minutes,  and  seconds  will  equal 
respectively  the  numbers  representing  the  difference  in  degrees,  min- 
utes, and  seconds  of  longitude.  Therefore,  the  difference  in  longitude 
is  35°  12'  15". 


204  LONGITUDE   AND   TIME. 

2.  The  difference  in  time  between  two  places  is  3  hr. 
16  min.  23  sec.     What  is  their  difference  in  longitude  ? 

3.  The   difference  in  time  between  Boston   and  New 
Orleans  is  1  hr.  16  min.  14  sec.     What  is  their  difference  in 
longitude  ? 

4.  The  difference  in  time  between  New  York  and  St. 
Louis  is  1  hr.  2  min.  20  sec.     What  is  the  difference  in 
their  longitude  ? 

5.  Two  persons  observed  the  occultation  of  a  certain 
star  by  the  moon,  one  seeing  it  at  9  P.M.,  and  the  other  at 
10^  P.M.     What  was  the  difference  in  their  longitude  ? 

6.  The  difference  in  time  between  Savannah,  Ga.,  and 
Portland,  Me.,  is  43  min.  32  sec.     What  is  their  difference 
in  longitude  ? 

7.  The  difference  in  time  between  London  and  New 
York  is  4  hr.  55  min.  37f  sec.     What  is  their  difference  in 
longitude  ? 

8.  When  it  is  12  o'clock  M.  at  Eochester,  N.Y.,  it  is  9 
hr.  1  min.  47  sec.  A.M.  at  San  Francisco.     The  longitude  of 
Eochester  is  77°  51'  west  from  Greenwich.     What  is  the 
longitude  of  San  Francisco  ? 

9.  When  it  is  noon  at  Greenwich  it  is  6  hr.  52  min.  40 
sec.  A.M.  at  Harrisburg,  Penn.     What  is  the  longitude  of 
Harrisburg  ? 

10.  A  traveler  found  on  arriving  at  his  destination  that 
his  watch  was  1  hr.  35  min.  too  slow.     In  which  direction 
had  he  been  traveling  ?     How  far  had  he  traveled  ? 

11.  When  it  is  noon  at  Philadelphia  it  is  10  min.  past  5 
o'clock  P.M.  at  Paris.     What  is  the  longitude  of  Paris,  the 
longitude  of  Philadelphia  being  75°  10'  ? 


PEACTICAL   MEASUEEMENTS. 


Acute  Angle 


Obtuse  Anglo 


242.  The  method  of  computing  the  area  of  a  rectangle 
and  a  square  was  learned  in  Art.  214,  but  there  are  other 
surfaces  whose  area  can  be  readily  found. 

243.  When  a  straight  line  meets  another 
straight  line  forming  two  equal  angles,  each 

angle  is  called  a  Right  Angle.  TWO  night  Augies 

When  two  lines  form  right  angles,  they  are  said  to 
be  perpendicular  to  each  other. 

244.  An  angle  smaller  than  a  right  angle 
is  called  an  Acute  Angle. 

245.  An  angle  larger  than  a  right  angle  is 
called  an  Obtuse  Angle. 

246.  Lines  which  are  equidistant  through-   

out  their  entire  length  are  called   Parallel     parallel  Lines 
Lines. 

247.  A  figure  having  four  straight  sides       /  / 
and  its  opposite  sides  parallel  is   called  a    / / 

Parallelogram.  Parallelogram 

1.  "When  the  angles  of  a  parallelogram  are  right 
angles,  it  is  called  a  Rectangle. 

2.  The  side  upon  which  a  figure  is  assumed  to 
stand  is  called  the  Base. 

3.  The  perpendicular  distance  between  the  base 
of  a  figure  and  the  highest  point  opposite  it  is  the 
Altitude. 

4.  The  straight  line  joining  the  opposite  angles  of  -~| 
a  parallelogram  is  called  its  Diagonal. 

205 


Rectangle 


206  PRACTICAL  MEASUREMENTS. 

WRITTEN   EXERCISES. 

248.    The  measurement  of  rectangles.     (See  §  214.) 
Find  the  area  of  a  rectangular  figure 

1.  15  rd.  by!2rd.  5.  37.5  yd.  by  8.3  yd. 

2.  36  rd.  by  24  rd.  6.  367  in.  by*4.12  in. 

3.  45  ft.  by  36  ft.  7.  384  ft.  by  21.6  ft. 

4.  400  ft.  by  80  ft.  8.  81.2  mi.  by  53.2  mi. 

9.    A  gable  roof  was  43  ft.  by  26.     How  many  square 
feet  of  tin  will  be  required  to  cover  it  ? 

10.  A  lot  was  18  rods  long  and  8  rods  wide.     What  part 
of  an  acre  did  it  contain  ? 

11.  What  was  the  value  of  the  above  lot  at  $342.50  per 
acre? 

12.  How  many  acres  are  there  in  a  square  farm,  each  of 
whose  sides  is  20  chains  ? 

13.  A  rectangular  piece  of  land  is  160  rods  long  and  120 
rods  wide.     How  many  acres  does  it  contain  ? 

14.  A  man  bought  a  rectangular  farm  40  ch.  long  and 
35  ch.  wide,  at  $  85  an  acre.     What  did  the  farm  cost  ? 

15.  The  area  of  a  certain  rectangular  garden  is  840  square 
yards,  and  its  length  is  35  yards.     How  wide  is  it  ? 

16.  A  rectangular  field  containing  8  acres  is  32  rods  wide. 
How  long  is  it  ? 

17.  A  rectangular  mirror  has  an  area  of  2520  sq.  in.     If 
its  width  is  3£  ft.,  what  is  its  length  ? 

18.  A  city  lot  containing  1610  sq.  yd.  has  a  front  of 
801  ft.     What  is  its  depth  ? 

19.  A  rectangular  farm  containing  100  acres  is  80  rods 
wide.     How  long  is  it  ? 


PARALLELOGRAMS.  207 

249.  The  measurement  of  parallelograms. 

It  is  apparent  that  the  parallelogram  ABCD  = 
the  rectangle  EFCD ;  that  the  base  AB  =  the  base       / 
EF,  and  that  the  altitude  of  each  is  DE.     Hence  a     / 
parallelogram  is  equivalent  to  a  rectangle  having  the    £- 
same  base  and  altitude. 

250.  PRINCIPLE.  —  The  area  of  a  parallelogram  is  equal 
to  the  product  of  the  numbers  expressing  its  base  and  altitude. 

The  base  and  altitude  must  be  expressed  in  units  of  the  same  de- 
nomination ;  the  product  will  then  express  the  area  in  square  units  of 
the  same  name. 

Find  the  area  of  the  following  parallelograms : 

1.  Base  46  ft.,  alt.  10  ft.         5.   Base  388  ft.,  alt.  125  ft. 

2.  Base  52  ft.,  alt.  12  ft.         6.   Base  175  ft.,  alt.  5  rd. 

3.  Base  47  ft.,  alt.  15  ft.         7.    Base  12  rd.,  alt.  45  yd. 

4.  Base  265  ft.,  alt.  119  ft.     8.   Base  275  rd.,  alt.  170  rd. 
9.   The  area  of  a  parallelogram  is  1628  sq.  ft.     Its  length 

is  74  ft.     What  is  its  altitude  ? 

10.  The  area  of  a  parallelogram  is  3404  sq.  ft.     It  has  an 
altitude  of  37  feet.     What  is  its  length  ? 

11.  I  have  a  lot  in  the  form  of  a  parallelogram  contain- 
ing one  acre.     The  distance  between  two  of  its  parallel  sides 
is  12  rods.     What  is  its  length  ? 

12.  The  area  of  a  field  is  3J  acres.     It  is  in  the  form  of 
a  parallelogram,  and  its  length  is  80  rods.     How  wide  is  it? 

13.  There  is  a  farm  in  the  form  of  a  parallelogram  con- 
taining 132  acres.     The  perpendicular  distance  between  the 
sides  is  132  rods.     What  ia  its  length  ? 

251.  To  find  the  area  of  a  triangle. 

252.  A  figure  having  three  sides  and  three 
angles  is  called  a  Triangle. 

The   point  where   the   sides  which  form  an 
angle  meet  is  called  the  Vertex. 


208  PRACTICAL   MEASUREMENTS. 

We  have  just  learned  that  the  area  of  a  paral- 
lelogram is  equal  to  the  product  of  the  numbers 
expressing  its  base  and  altitude.  It  is  evident 
that  a  diagonal  of  a  parallelogram  divides  it  into 
two  equal  triangles.  Hence, 

253.  PRINCIPLE.  —  The  area  of  a  triangle  is  one  half  the 
product  of  the  numbers  expressing  its  base  and  altitude. 

Find  the  area  of  the  following  triangles : 

1.  Base  30  ft.,  alt.  12  ft.         4.   Base  54  ft.,  alt.  43  ft. 

2.  Base  45  ft.,  alt.  30  ft.         5.   Base  67  ft.,  alt.  55  ft. 

3.  Base  37  ft.,  alt.  28  ft.         6.   Base  40  ft.,  alt.  25  ft. 

7.  The  base  of  a  triangular  field  is  360  yards,  and  the 
altitude  is  615  feet.     How  many  acres  does  it  contain  ? 

8.  What  will  be  the  cost  of  a  triangular  piece  of  land 
whose  base  is  18.36  ch.,  and  the  altitude  10.54  ch.,  at  $  70 
per  acre  ? 

9.  How  many  square  feet  of  boards  will  be  required  to 
cover  the  gables  of  a  house  that  is  28  ft.  wide,  the  ridge  of 
the  roof  of  the  house  being  13  ft.  higher  than  the  foot  of 
the  rafters  ? 

254.  To  find  the  area  of  a  trapezoid. 

255.  A  figure  having  four  sides,  two  of  which  are  parallel, 
is  called  a  Trapezoid. 

It  is  evident  that  any  trapezoid  may  be  divided  into  two  triangles 
by  a  line ;  as,  AC.     The  area  of  one  triangle  is  the 
product  of  one  half  the  length  of  one  of  the  parallel 
sides,  as  AD,  multiplied  by  the  altitude  CE,  and 
the  area  of  the  other  triangle  is  the  product  of  one 
half  the  length  of  the  other  parallel  side,  as  CB,     A  ' 
multiplied  by  the  altitude  CE.     Therefore, 


TRAPEZOIDS.  209 

256.  PRINCIPLE.  —  The  area  of  a  trapezoid  is  equal  to  the 
length  of  one  half  the  sum  of  the  parallel  sides  multiplied  by 
the  altitude. 

Find  the  area  of  the  following  trapezoids : 

1.  Altitude  10  ft.,  parallel  sides  15  ft.  and  11  ft. 

2.  Altitude  12  ft.,  parallel  sides  16  ft.  and  14  ft. 

3.  Altitude  18  ft.,  parallel  sides  20  ft.  and  18  ft. 

4.  Altitude  25  ft.,  parallel  sides  28  ft.  and  23  ft. 

5.  Altitude  46  ft.,  parallel  sides  54  ft.  and  39  ft. 

6.  There  is  a  field  in  the  form  of  a  trapezoid  whose 
altitude  is  32  rd.,  and  whose  parallel  sides  are  42  rd.  and 
50  rd.  long,  respectively.     How  many  acres  are  there  in  the 
field? 

7.  A  field  in  the  form  of  a  trapezoid,  which  has  an  alti- 
tude of  40  rd.  and  whose  parallel  sides  are  52  rd.  and  58 
rd.,  respectively,  contains  how  many  acres  ? 

8.  A  farm  in  the  form  of  a  trapezoid  has  its  parallel 
sides  64  ch.  and  76  ch.  in  length.     The  perpendicular  dis- 
tance between  them  is  192  rods.     How  large  is  the  farm  ? 

9.  There  is  a  field  in  the  form  of  a  trapezoid  which  has 
an  altitude  of  180  rd.     The  parallel  sides  are  96  rd.  and  108 
rd.  long,  respectively.     How  many  acres  are  there  in  the 
field? 

10.  One  side  of  a  field  is  38  chains  long,  the  side  parallel  to 
it  is  28  chains  long,  and  the  perpendicular  distance  between 
them  is  25  chains.     How  many  acres  are  there  in  the  field  ? 

11.  What  are  the  square  contents  of  a  walk,  in  the  form 
of  a  trapezoid,  20  ft.  long  and  3  ft.  wide  at  one  end  and 
5  ft.  at  the  other? 

STAND.  AR.  •=«-  14 


210 


PRACTICAL  MEASUREMENTS. 


257.   To  find  the  circumference  or  diameter  of  a  circle. 

A  plane  figure,  bounded  by  a  curved  line,  every  point  of 
which  is  equally  distant  from  a  point  within,  called  the 
center,  is  a  Circle. 


1.  The  line  which  bounds  a  circle  is  called  its 
Circumference. 

2.  The  straight  line  drawn  from  the  center  of  the 
circle  to  the  circumference  is  called  the  Radius. 

3.  A  straight  line  drawn  through  the  center  of  a 
circle,  terminating  at  both  ends  in  the  circumfer- 
ence, is  called  the  Diameter. 

4.  A  radius  is  one  half  a  diameter. 


Circle 


258.  PRINCIPLE.  — The  circumference  of  a  circle  is  about 
3}  times  the  diameter;  or  more  accurately,  3.1416  times  the 
diameter. 

Find  the  approximate  circumferences  of  circles  having 
the  following  diameters : 

1.  35ft.       3.  63ft.       5.  84ft.       7.     98ft.        9.  126ft. 

2.  77  ft.       4.  49  ft.       6.  56  ft.       8.  105  ft.      10.  168  ft. 

Find  the  more  accurate  circumferences  of  the  following 
circles : 


11.  Diameter  15  ft. 

12.  Diameter  20  ft. 

13.  Diameter  24  ft. 

14.  Diameter  19  ft. 

15.  Diameter  29  ft. 


16.  Diameter  35  ft. 

17.  Eadius  8  ft. 

18.  Eadius  12£  ft. 

19.  Kadius  15|  ft. 

20.  Eadius  22|  ft. 


21.   What  is  the  diameter  of  a  circle  whose  circumfer- 
ence is  37.6992  ft. 

SOLUTION.  — 37.6992  ft.  -4-  3.1416  =  12  ft.,  the  diameter. 


CIRCLES. 


211 


Find  the  diameters  of  circles  having  the  following  cir- 
cumferences : 

22.  14.3  ft.         24.  318  ft.         26.  670  ft.  28.  1200  rd. 

23.  164  ft.          25.  426  ft.         27.  955.5  rd.       29.  1676  rd. 

259.  To  find  the  area  of  a  circle. 

From  the  accompanying  figure,  it  is 
evident  that  a  circle  may  be  regarded  as 
composed  of  a  large  number  of  triangles, 
the  sum  of  whose  bases  forms  the  circum- 
ference of  the  circle,  and  whose  altitude 
is  the  radius  of  the  circle.  Hencs, 

260.  PRINCIPLE.  —  Tlie  area  of  a  circle  is  equal  to  the 
circumference,  multiplied  by  half  tJie  radius. 

Find  the  area  of  the  following  circles  : 

1.  Circum,  50  ft.,  diam.  15.915  ft.       6.  Circum.  869  rd. 

2.  Circum.  60  ft.,  diam.  19.098  ft.       7.  Circum.  728  rd. 

3.  Circum.  37.6992  ft.,  diam.  12  ft.     8.  Diam.  240  rd. 

4.  Circum.  314.16  ft.,  diam.  100  ft.     9.  Diam.  125  ft. 

5.  Circum.  640  ft.,  diam.  203.7  ft.     10.  Diam.  364  rd. 

11.  What  is  the  area  of  a  circular  field,  whose  circum- 
ference is  320  rd.,  and  whose  diameter  is  101.856  rd.? 

12.  The  circumference  of  a  circular  field  is  436  rd.    How 
many  acres  does  it  contain  ? 

261.  To  find  the  cost  of  plastering,  painting,   and  kalso- 
mining. 

Plastering,  painting,  and  kalsomining  are  usually  computed 
by  the  square  yard. 

Allowances  are  sometimes  made  for  the  whole  or  part  of  the  area 
of  openings,  and  for  baseboards,  but  custom  varies  so  greatly  that  a 
written  contract  regarding  the  allowances  should  be  made  to  avoid 
complications  at  the  time  of  settlement.  In  the  examples  given,  the 
plastering  is  considered  to  extend  only  to  the  baseboard. 


212 


PRACTICAL  MEASUREMENTS. 


------  ifr-  ----- 

RECEPTION 


1.  Find  the  cost  of  plastering  a  room  18  ft.  by  16  ft.  and 
12  ft.  high,  at  35^  per  sq.  yd.  —  no  allowance  for  openings. 

EXPLANATION.  —  This  dia- 
gram shows  the  plan  of  the 
rooms  on  the  first  floor  of  a 

!4»    SITTING  p3    LIBRARY     ui       two-story  brick  house.      The 

?     ROOM  dimensions  are  as  follows : 

House.  —40  ft.  long,  31  ft. 
wide,  and  25  feet  high. 

Rooms.  —  As  shown  in  dia- 
gram, and  of  the  uniform  height 
of  10  feet. 

Doors.  —  Front,  5  ft.  by 
8  \  ft.  ;  the  doors  between 
the  parlor  and  library,  and  between  the  reception-room  and  sitting, 
room,  each  6  ft.  by  8  ft. ;  all  others  3  ft.  by  7  ft. 

Windows.  —Front  3£  ft.  by  8  ft.  ;  all  others  3  ft.  by  6  ft. 

Baseboards.  —  Uniformly  9  in.  wide. 

Second  Floor.  —  Similar  to  first  floor,  except  no  outside  doors. 

2.  Find  the  cost  of  plastering  the  reception-room,  walls, 
and  ceiling,  at  30^  a  sq.  yd.,  deducting  for  one  half  the  area 
of  the  openings. 

3.  What  will  it  cost  to  plaster  the  walls  and  ceiling  of 
the  sitting-room,  at  35^  a  sq.  yd.,  making  full  deduction  for 
openings  ? 

4.  What  will  be  the  cost  of  plastering  the  walls  and  ceil- 
ing of  the  library,  at  30^  a  sq.  yd.,  deducting  for  one  half 
the  area  of  the  openings  ? 

5.  What  will  be  the  cost  of  plastering  the  walls  and  ceil- 
ing of  the  parlor  on  the  terms  given  for  plastering  the 
library  ? 

6.  Find  the  cost  of  kalsomining  the  ceilings,  including 
the  hall,  at  8^  a  sq.  yd. 

7.  What  will  be  the  total  cost  of  painting  the  outside 
brick  work  of  the  house,  2  coats,  each  8-^  a  sq.  yd.,  deduct- 
ing for  doors  and  windows  ? 


CARPETING.  213 

262.   To  find  the  cost  of  carpeting. 

Carpets  are  commonly  either  1  yd.  or  f  yd.  in  width,  but 
matting,  oilcloth,  and  other  materials  are  of  various  widths. 

1.  In  matching  the  patterns  in  carpets  there  is  usually  some  waste. 

2.  Sometimes  carpets  are  necessarily  made  a  little  too  wide  and 
are  turned  under,  consequently  in  computing  the  cost  of  carpets  the 
number  of  strips  of  carpet  must  be  found. 

3.  When  borders  are  put  around  carpets  the  corners  must  be  counted 
twice  because  one  half  of  each  corner  is  wasted  in  making. 

1.  A  room  36  ft.  long  and  18  ft.  wide  is  carpeted  with 
ingrain  carpet  1  yd.  wide  without  waste  in  matching.    What 
will  be  the  cost  of  the  carpet  at  $  .85  per  lineal  yard  ? 

2.  If  the  floor  is  covered  by  paper  lining  at  10^  per  sq. 
yd.,  and  5  cents  per  lineal  yd.  is  charged  for  laying  the  car- 
pet, what  will  be  the  entire  cost  of  carpeting  the  room  ? 

3.  How  many  yards  of  carpet,  a  yard  wide,  will  be  re- 
quired for  a  room  24  ft.  long  and  17  ft.  wide,  if  the  strips 
run  lengthwise  and  there  is  no  waste  in  matching  ? 

4.  What  will  the  carpet  cost,  at  $1.75  per  lineal  yard, 
and  what  will  be  the  entire  cost  if  the  floor  is  first  covered 
with  paper  lining?  at  9^  per  sq.  yard  ? 

5.  Find  the  cost  of  a  carpet  27  inches  wide,  at  $  1.60  per 
lineal  yard,  for  a  room  15  ft.  long  and  13^  ft.  wide,  if  the 
strips  run  lengthwise.      Find  the  cost,  if  the  strips  run 
across  the  room. 

6.  How  many  yards  of  carpet  27  in.  wide  will  be  required 
for  a  room  18  ft.  long  and  16  ft.  wide,  if  the  strips  run 
lengthwise  and  there  is  an  average  waste  of  £  of  a  yard  per 
strip  in  matching  the  pattern  ?     What  will  be  the  cost  of 
the  carpet  at  $  1.85  per  lineal  yard  ? 

7.  Find  the  cost  of  a  carpet  f  of  a  yard  wide,  at  $  1.62^- 
per  lineal  yard,  for  a  room  19^-  ft.  long  and  13^  ft.  wide,  if 
the  strips  run  lengthwise,  and  if  there  is  an  average  waste 
of  -|-  of  a  yard  per  strip  in  matching  the  pattern. 


214  PRACTICAL  MEASUREMENTS. 

8.  What  will  be  the  cost  of  a  rug  2\  yd.  by  3  yd.,  at 
$  1.25  per  sq.  yd.,  with  a  border  f  of  a  yard  wide,  in  addi- 
tion, at  $  .75  per  lineal  yard  ? 

263.  To  find  the  cost  of  papering. 

Wall  paper  is  sold  by  the  roll,  and  in  computations  any 
part  of  a  roll  is  considered  a  whole  roll. 

A  roll  is  8  yd.  long  and  18  in.  wide,  unless  otherwise 
specified. 

1.  The  width  of  a  roll  given  above  is  that  commonly  used  in  America, 
but  imported  papers  differ  as  to  the  width  and  the  length  of  the  roll. 

2.  Paper  is  often  put  up  in  double  rolls,  16  yd.  long,  so  as  to  econo- 
mize the  waste  in  cutting.     Double  rolls  are  counted  as  2  rolls  each. 

3.  Borders  or  friezes  are  sold  by  the  yard,  and  vary  in  width  from 
3  in.  upward. 

It  is  rarely  possible  to  find  the  exact  cost  of  papering  a 
room,  but  the  following  process  will  approximate  accuracy  : 

RULE.  —  Measure  the  entire  distance  around  the  room  in 
yards.  The  number  of  strips  will  be  double  the  number  of  yards. 

Find  then  how  many  strips  can  be  cut  from  a  roll,  and 
divide  the  number  of  strips  required  to  go  around  the  room  by 
the  number  that  can  be  cut  from  a  roll.  The  quotient  will  be 
the  number  of  rolls. 

1.  How  many  rolls  of  paper,  8  yd.  long  and  18  in.  wide, 
will  be  required  to  paper  the  walls  of  a  room  18  ft.  long, 
15  ft.  wide,  and  having  a  height  of  8  ft.  from  the  baseboard, 
which  is  9  in.  high,  to  the  ceiling,  allowing  for  one  door  3  ft. 
by  7  ft.,  and  for  two  windows,  each  3  ft.  by  6  ft.  ? 

2.  Find  the  number  of  double  rolls  of  paper  required  to 
paper  the  walls  of  the  reception-room  described  on  page  212, 
making  no  deductions  for  doors  or  windows. 

3.  Find  the  number  of  double  rolls  of  paper  required  for 
the  walls  of  the  sitting-room  described  on  page  212. 


PAVING,  BRICK  AND  STONE   WORK.  215 

4.  Find  the  number  of  double  rolls  of  paper  required  for 
the  walls  of  the  parlor  represented  in  the  same  diagram. 

5.  How  many  double  rolls  of  paper  will  be  required  for 
the  walls  of  the  hall  shown  in  the  diagram,  making  full 
deductions  for  the  openings  ? 

6.  What  will  be  the  cost  of  papering  the  parlor  repre- 
sented on  page  212,  at  $1.20  per  roll  for  paper  and  putting 
it  on,  with  a  border  or  a  frieze  18  in.  wide,  at  $  .50  per  yard 
for  the  border  and  putting  it  on,  allowing  one  half  the 
area  of  the  openings  ? 

264.  To  compute  the  expense  of  paving. 
Find  the  cost  of  paving : 

1.  A  sidewalk  5  ft.  wide,  40  ft.  long,  at  25^  per  sq.  ft. 

2.  A  sidewalk  36  ft.  long,  5  ft.  6  in.  wide,  with  brick,  at 
$  1.35  a  sq.  yd.,  the  bricks  to  be  laid  on  edge. 

3.  A  courtyard  20  ft.  by  24  ft.  9  in.,  with  bricks  laid  flat 
in  sand,  at  80^  a  sq.  yd. 

4.  How  much  less  would  it  cost  to  make  a  brick  side- 
walk 4£  ft.  wide  and  260  ft.  long,  at  $  1.08  a  sq.  yd.,  than 
to  lay  a  stone  walk  of  the  same  dimensions,  at  22^  a  sq.  ft. 

265.  Brick  and  stone  work. 

Stone  work  is  commonly  estimated  by  the  perch. 

1.  A  perch  of  stone  work  is  16|  ft.  long,  \\  ft.  wide,  and  1  ft.  thick, 
or  24  £  cubic  feet. 

2.  It  is  customary  in  many  places  to  reckon  masonry  by  the  cubic 
foot  instead  of  by  the  perch. 

3.  In  estimating  the  work  of  laying  stone  and  brick  the  corners  are 
commonly  doubled,  but  not  in  computing  the  quantity  of  material  used. 

4.  Usually  a  deduction  is  made  for  one  half  of  the  openings. 

5.  A  cubic  foot  of  wall  contains  about  22  common  bricks ;  hence, 
a  wall  contains  22  bricks  for  each  square  foot  of  face,  if  12  in.  thick, 
and  7  bricks  more  per  square  foot  for  each  additional  4  in.  in  thick- 
ness ;  but  as  bricks  vary,  computation  must  be  made  for  each  size. 


216  PRACTICAL  MEASUREMENTS. 

1.  How  many  common  bricks  are  there  in  a  wall  20  ft. 
long,  6  ft.  high,  and  12  in.  thick  ? 

2.  What  will  it  cost  to  build  a  wall  38  ft.  long,  7  ft.  high, 
and  16  in.  wide,  if  built  of  common  bricks  at  an  expense  of 
$  11  per  M,  allowance  being  made  for  a  gate  10  ft.  wide  ? 

3.  How  many  perches  of  stone  will  be  required  to  build 
the  walls  of  a  cellar  36  ft.  long  and  24  ft.  wide,  the  walls 
to  be  8  ft.  high  and  18  in.  thick,  deducting  96  cu.  ft.  for 
openings  ? 

4.  How  much  should  a  mason  receive  for  building  the 
walls  of  the  above  cellar,  if  he  charges  $  1.60  a  perch  for  his 
labor  ? 

5.  At  27^  per  cubic  foot,  how  much  must  be  paid  for 
building  the  walls  of  a  cellar  that  is  44  ft.  long  and  38  ft. 
wide ;  the  walls  to  be  8  ft.  high  and  2  ft.  thick,  no  allow- 
ance being  made  for  openings  ? 

6.  What  will  it  cost  to  build  of  common  bricks  a  house 
32  ft.  long,  30  ft.  wide,  and  25  ft.  high,  the  walls  being 
16  in.  thick,  when  brick  costs  $  8.50  per  M,  and  laying  the 
brick  is  paid  for  at  the  rate  of  $2  per  M,  making  full 
deductions  for  two  doors,  each  7  ft.  by  3£  ft.,  and  12  win- 
dows, each  6  ft.  by  3  ft.  ? 

266.  To  find  the  quantity  of  wood. 

A  cord  of  wood  or  stone  is  a  quantity  of  material  8  ft. 
long,  4  ft.  wide,  and  4  ft.  thick,  or  128  cu.  ft. 

How  many  cords  of  wood  are  there  in  the  following : 

1.  In  a  pile  18  ft.  long,  4  ft.  wide,  and  6  ft.  high  ? 

2.  In  a  pile  23  ft.  long,  4  ft.  wider  and  4J-  ft.  high? 

3.  In  a  pile  28  ft.  long,  3£  ft.  wide,  and  4  ft.  high  ? 

4.  In  a  pile  15  ft.  long,  5  ft.  wide,  and  7  ft.  high  ? 


LUMBER.  217 

5.  A  man  bought  a  pile  of  wood  9  ft.  long,  4  ft.  wide, 
and  4£  ft.  high,  at  $3.50  per  cord.     How  much  did  it  cost 
him? 

6.  What  will  be  the  cost  of  a  pile  of  stones  25  ft.  long, 
4  ft.  wide,  and  5  ft.  high,  at  $3.80  per  cord  ? 

7.  A  man  bought  5  loads  of  wood,  each  load  7  ft.  long, 
3J  ft.  wide,  and  4  ft.  high.     What  did  it  all  cost  at  $3.75 
a  cord  ? 

8.  How  many  cords  of  stone  are  there  in  a  pile  75  ft. 
long,  4  ft.  wide,  and  5|  ft.  high  ? 

9.  How  many  cords  of  wood  can  be  placed  in  a  shed 
24  ft.  long,  20  ft.  wide,  and  16  ft.  high  ? 

267.   To  measure  lumber. 

In  measuring  lumber,  boards  1  inch  thick  or  less  are  esti- 
mated by  the  square  foot  of  surface. 

Thus,  a  board  1  foot  wide  and  15  feet  long  would  contain  15  square 
feet,  or  15  feet  board  measure,  if  the  board  were  1  inch  or  less  in 
thickness. 

When  lumber  is  more  than  1  inch  in  thickness,  the  number 
of  feet  board  measure  is  obtained  by  multiplying  the  length 
in  feet  by  the  breadth  in  feet,  and  this  product  by  the  number 
of  inches  in  thickness. 

Thus,  the  number  of  feet  board  measure  in  a  timber  18  feet  long, 
15  in.  wide,  and  2£  in.  thick  is  obtained  as  follows: 
18  ft.  x  H  x  2£  =  50f  ft. 

The  width  of  a  board  that  tapers  uniformly  is  measured  at  the 
middle,  thus  securing  the  average  width. 

The  average  width  is  one  half  the  sum  of  the  two  ends. 

How  many  feet  are  there  in  the  following  boards : 

1.  12  ft.  long,  15  in.  wide.        3.   16  ft.  long,  18  in.  wide. 

2.  15  ft.  long,  14  in.  wide.        4.   20  ft.  long,  9  in.  wide. 


218  PRACTICAL  MEASUREMENTS. 

How  many  board  feet  are  there  in  the  following  timbers? 

5.  30  ft.  by  15  in.,  and  4  in.  thick. 

6.  28  ft.  by  14  in.,  and  6  in.  thick. 

7.  24  ft.  long  and  9  in.  square. 

8.  What  will  be  the  cost  of  25  joists  20  ft.  long,  16  in. 
wide,  and  3£  in.  thick,  at  $  15  per  M  ? 

9.  What  will  be  the  cost  of  20  planks  18  ft.  long,  16  in. 
wide,  and  2£  in.  thick,  at  $18  per  M  ? 

10.  What  will  be  the  cost  of  a  board  20  feet  long,  22  in. 
wide  at  one  end  and  16  in.  at  the  other,  and  1^-  in.  thick, 
at  $25  per  M  ? 

268.  To  find  the  capacity  of  bins,  etc. 

2150.42  cubic  inches  =  1  bushel. 

Find  the  contents  in  bushels  of  the  following : 

1.  A  box  3  ft.  long,  2  ft.  wide,  and  2%  ft.  high. 

2.  A  box  4  ft.  long,  2£  ft.  wide,  and  3  ft.  high. 

3.  A  box  5  ft.  long,  3  ft.  wide,  and  4J  ft.  high. 

4.  How  many  bushels  of  grain  will  a  bin  hold  that  is  5 
ft.  long,  3£  feet  wide,  and  6  ft.  high  ? 

5.  I  wish  to  make  a  bin  5  ft.  square  that  will  contain 
100  bushels  of  grain.     How  high  must  the  bin  be  made  ? 

6.  A  wagon-box  is  11  ft.  long,  3£  ft.  wide,  and  2£  ft. 
deep.     How  many  bushels  of  grain  will  fill  it  even  full  ? 

269.  To  find  the  capacity  of  cisterns. 

231  cubic  inches  =  1  gallon. 
31  £  gallons  =  1  barrel. 

Find  the  contents  of  the  following : 

1.  A  tank  3  ft.  by  4  ft.,  and  5  ft.  deep. 

2.  A  tank  3£  ft.  by  4  ft.,  and  4  ft.  deep. 


APPROXIMATE  MEASUREMENTS.  219 

3.  A  tank  4  ft.  by  4J-  ft.,  and  5f  ft.  deep. 

4.  A  tank  5J  ft.  by  6  ft.,  and  6£  ft.  deep. 

5.  How  many  gallons  will  a  cistern  contain  that  is  5  ft. 
square  and  8  ft.  deep  ? 

6.  How  many  barrels  of  water  will  a  circular  cistern  6  ft. 
in  diameter  and  8  ft.  deep  hold  ? 

SUGGESTION.  —  The  area  of  the  bottom  may  be  found  by  Art.  260. 
That  multiplied  by  the  depth  will  be  the  volume. 

7.  How  many  barrels  would  be  required  to  fill  a  circular 
cistern  8  ft.  in  diameter  and  16  ft.  deep  ? 

270.   Approximate  measurements. 

1.  A  bushel  is  nearly  1£  cu.ft.     Hence  f  of  the  number  of  cubic 
feet  is  very  nearly  the  number  of  bushels. 

2.  A  cubic  foot  of  any  liquid  contains  nearly  7|  gallons ;  a  barrel 
of  31|  gallons  about  4$-  cu.  ft. 

3.  A  ton  of  fine  hay  in  a  well-settled  large  stack,  or  mow,  is  about 
450  cu.  ft.     A  ton  of  clover  hay  is  about  550  cu.  ft. 

A  ton  of  Lehigh  stove  coal  is  about  34 1  cu.  ft. 

A  ton  of  Schuylkill  white  ash  stove  coal  is  about  35  cu.  ft. 

A  ton  of  red  ash  stove  coal  is  about  36  cu.  ft. 

1.  About  how  many  bushels  of  grain  will  a  box  4  ft.  long, 
3  ft.  wide,  and  5  ft.  high  hold  ? 

2.  A  watering  trough  is  8  ft.  long,  14  in.  wide,  and  12 
in.  deep.     About  how  many  gallons  will  it  hold  ? 

3.  A  tank  6  ft.  square  and  8  ft.  deep  will  hold  about  how 
many  barrels  ? 

4.  About  how  many  tons  of  fine  hay  can  be  packed  in  a 
mow  20  ft.  by  18  ft.,  and  8  ft.  high  ? 

5.  How  many  tons,  approximately,  of  clover  hay  are 
there  in  a  stack  20  ft.  long,  15  ft.  wide,  and  12  ft.  high  ? 

6.  How  many  tons  of  Lehigh  stave  coal  can  be  put  in  a 
bin  12  ft.  long,  8  ft.  wide,  and  6  ft.  deep  ?    How  many  tons 
of  Schuylkill  white  ash  stove  coal?     How  many  tons  of 
red  ash  stove  coal  ? 


GENERAL  REVIEW   EXERCISES. 


ORAL  EXERCISES. 

271.   1.   If  3  pears  cost  5f  cents,  what  will  5  pears  cost 
at  the  same  rate  ?     What  will  8  pears  cost  ? 

2.  If  5  peaches  cost  6-J  cents,  what  will  9  peaches  cost 
at  the  same  rate  ?     What  will  20  cost  ? 

3.  How  much  will  7  oranges  cost  at  the  rate  of  6  oranges 
for  6f  cents  ? 

4.  If  8  books  are  worth  $  11|,  what  are  10  books  worth 
at  the  same  rate  ?     What  are  20  worth  at  the  same  rate  ? 

5.  If  9  ducks  cost  $  6f,  what  will  14  cost  ? 

6.  If  4  pigs  cost  $  13J,  how  much  will  12  pigs  cost  at 
the  same  rate  ?     How  much  will  16  cost  ? 

7.  How  many  sheep  can  be  bought  for  $  56  when  4  sheep 
cost  $  14  ? 

8.  A  man  gave  $  72  for  potatoes,  at  the  rate  of  $  8  for 
9^  bushels.     How  many  bushels  did  he  buy  ? 

9.  How  much  must  be  paid  for  22  yards  of  ribbon  at 
the  rate  of  5  yards  for  7-^  dimes  ? 

10.  What  must  be  paid  for  8  shovels  when  5  shovels  are 
sold  for  $3J? 

11.  If  £  of  a  barrel  of  flour  costs  $5,  what  will  7  barrels 
cost  ?     What  will  10  barrels  cost  ? 

12.  Mr.  A  gave  J  of  his  month's  salary  for  a  coat  and  \ 
for  board,  and  had  $  33  left.     What  was  his  salary  ? 

13.  One  third  of  the  trees  in  an  orchard  bear  apples,  \ 
peaches,  and  all  the  other  trees,  which  are  30,  bear  plums. 
How  many  trees  are  there  in  the  orchard  ? 

220 


ORAL  EXERCISES.  221 

14.  If  f  of  a  ship  is  worth  $  15000,  how  much  is  £  of  it 
worth  ? 

15.  Harry  spent  80  cents  for  a  book,  which  was  -]-§•  of  his 
money ;  with  the  remainder  he  bought  oranges  at  2  cents 
apiece.     How  many  oranges  did  he  buy  ? 

16.  A  farmer  bought  a  quantity  of  goods  and  paid  f  30 
cash,  which  was  f  of  the  value  of  the  goods.     How  many 
cords  of  wood,  at  $  3-J-  per  cord,  will  it  take  to  pay  for  the 
remainder  ? 

17.  If  7  men  can  do  a  piece  of  work  in  10^  days,  how 
long  will  it  take  9  men  to  do  the  same  work  ? 

18.  It  required  6  days  for  20  men  to  load  a  vessel.    How 
many  men  would  be  required  to  load  it  in  2^-  days  ? 

19.  If  it  costs  $  40  to  support  a  family  of  8  persons  for 
2-J-  weeks,  what  will  it  cost  to  support  11  persons  4  weeks  ? 

20.  Mr.  B  paid  %  of  his  money  for  a  horse,  \  of  the  re- 
mainder for  a  suit  of  clothes,  \  of  the  remainder  for  provi- 
sions, and  had  $  60  left.    How  much  money  had  he  at  first  ? 

21.  Three  men  engage  to  husk  a  field  of  corn.     The  first 
can  do  it  in  10  days,  the  second  in  12,  and  the  third  in  15 
days.     In  what  time  can  they  do  it  together  ? 

22.  One  half  of  Charles's  money  equals  £  of  Henry 's,  and 
Charles  has  $  12  more  than  Henry.     How  much  has  each  ? 

23.  If  a  pole  12  ft.  long  casts  a  shadow  17  ft.  long,  what 
is  the  length  of  a  pole  which  casts  a  shadow  85  ft.  long  at 
the  same  time  ? 

24.  A  man  bought  a  horse  and  carriage  for  $400,  and 
the  horse  cost  f  as  much  as  the  carriage.     What  was  the 
cost  of  each  ? 

25.  At  a  certain  election  the  successful  candidate  had  a 
majority  of  100,  which  was  -^  of  all  the  votes  cast.     How 
many  votes  did  the  defeated  candidate  receive  ? 


222  GENERAL  REVIEW  EXERCISES. 

26.  A,  B,  and  C  together  have  $2700.     How  much  has 
each,  if  A  has  3^  times  as  much  as  B,  and  C  as  much  as  A 
and  B  together  ? 

27.  How  many  square  feet  are  there  in  the  surface  of  a 
cube  whose  dimensions  are  3  inches  ? 

28.  Three  men  can  do  a  piece  of  work  in  5  days.     The 
first  can  do  it  in  15  days,  and  the  second  can  do  it  in  20 
days.     How  long  will  it  take  the  third  to  do  it  ? 

29.  Three  men  bought  a  thrashing  machine  for  $560. 
The  first  paid  for  £  of  it,  the  second  for  ^  of  it,  and  the  third 
paid  for  the  rest.     What  should  the  third  receive  as  his 
share  of  the  profits,  if  they  gained  $  300  ? 

30.  A  lad  spent  on  July  4th  -1-  of  his  money  and  6  cents- 
more  for  firecrackers,  and  ^  of  it  and  4  cents  more  for  tor- 
pedoes.   If  that  was  all  the  money  he  had,  how  much  had 
he? 

SUGGESTION.  —  \  of  his  money  and  6  cents,  plus  £  of  it  and  4  cents,. 
is  f  of  it  and  10  cents,  which  was  equal  to  the  whole  of  his  money. 

31.  While  A  earns  $  3,  B  earns  $  4,  and  C  earns  $  5.    At 
the  end  of  a  certain  time  the  earnings  of  all  were  $60, 
How  much  had  each  earned  ? 

32.  John  bought  a  certain  number  of  oranges  at  the  rate- 
of  4  for  5  cents,  and  sold  them  at  the  rate  of  3  for  4  cents. 
He  gained  25  cents.     How  many  did  he  buy  ? 

33.  A  fox  is  90  rods  before  a  hound,  and  runs  12  rods 
while  the  hound  runs  13.     How  far  will  the  fox  run  before 
he  is  overtaken  ? 

34.  One  third  of  a  certain  number  is  12  more  than  J  of 
it.     What  is  the  number  ? 

35.  If  to  a  certain  number  you  add  \  of  itself  and  ^  of 
itself,  the  sum  will  be  87.     What  is  the  number  ? 


ORAL  EXERCISES.  223 

36.  An  octavo  book  contains  480  pages.      How  many 
reams  of  paper  will  it  require  to  print  an  edition  of  1600 
copies,  making  no  allowance  for  waste  ? 

37.  Two  men  divided  a  lot  of  wood  which  they  had  pur- 
chased together  for  $  45.     One  took  7  cords,  and  the  other 
took  8  cords.     What  ought  each  to  pay  ? 

38.  A  and  B  hired  a  pasture  for  $30.    A  put  in  4  horses 
for  5  weeks,  and  B  5  horses  for  6  weeks.     How  much  ought 
each  to  pay  ? 

39.  If  to  a  certain  number  you  add  17  more  than  £  of 
itself,  the  sum  will  be  82.     What  is  the  number  ? 

40.  R  and  W  can  do  a  piece  of  work  in  15  days.     If  E 
does  -f  as  much  as  W,  in  how  many  days  can  each  do  it 
alone  ? 

41.  A,  B,  and  C  own  720  acres  of  land.     How  much  has 
each  if  A  owns  3  times  as  much  as  C,  and  B  4  times  as  much 
as  C? 

42.  S,  T,  and  V  have  $580.     How  much  has  each  if  S 
has  |  as  much  as  V,  and  T  has  f  as  much  as  V  ? 

43.  If  3  men  or  5  boys  can  do  a  piece  of  work  in  20  days, 
how  long  will  it  take  4  men  and  10  boys  to  do  it? 

44.  A  certain  piece  of  work  can  be  done  by  5  men  or  by 
8  boys  in  12  days.     In  how  many  days  can  5  men  and  8 
boys  do  it  ? 

45.  A  farmer  put  his  grain  into  four  bins.    Into  the  first 
lie  put  |  of  it,  into  the  second  J,  into  the  third  £,  and  into 
the  fourth  56  bushels.    How  many  bushels  of  grain  had  he  ? 

46.  A  man  paid  $36  for  a  cow,  and  f  of  the  price  paid 
for  the  cow  was  -|  of  the  cost  of  a  horse.     What  did  the 
horse  cost  ? 

47.  A  man  in  trade  lost  f  of  the  money  he  invested,  after 
which  he  gained  $  700  j  he  then  had  $  3400.     What  was  his 
total  loss  ? 


224  GENERAL  REVIEW  EXERCISES. 

48.  I  paid  $  1.50  for  flour  at  the  rate  of  $6  a  barrel. 
How  many  pounds  did  I  get  ? 

49.  A  pole  whose  length  is  76  feet  was  broken  into  two 
parts,  so  that  £  of  the  first  part  was  equal  to  f  of  the  second 
part.     What  was  the  length  of  each  part  ? 

50.  A  man  owning  ^  of  a  vessel  sold  f  of  what  he  owned 
for  $  1200.    What  was  the  value  of  the  vessel  at  that  rate  ? 

51.  I  have  8  lots,  5  of  which  are  each  7  rods  square,  and 
the  others  contain  each  10  square  rods.     How  many  square 
rods  of  land  do  I  own  ? 

52.  How  many  square  inches  of  surface  has  a  cubical 
block  whose  dimensions  are  each  6  inches  ? 

WRITTEN   EXERCISES. 

272.     1.   If  36  sheep  are  worth  $95,  what  are  60  sheep 
worth  at  the  same  rate  ? 

2.  If  14  men  can  build  a  house  in  30  days,  how  long 
would  it  take  30  men  to  build  it  ? 

3.  If  48  rods  of  ditching  cost  $72,  what  will  142  rods 
cost  at  the  same  rate  ? 

4.  At  the  rate  of  840  miles  in  24  hours,  how  far  will  a 
train  of  cars  go  in  19  hours  ? 

5.  If  120  acres  of  land  produce  2520  bushels  of  wheat, 
how  many  bushels  will  160  acres  produce  ? 

6.  If  21  acres  produce  35  tons  of  hay,  how  many  acres 
will  it  require  to  produce  100  tons  ? 

7.  If  18f  yards  of  cloth  cost  $  38.50,  what  will  40|  yards 
of  the  same  cloth  cost  ? 

8.  If  36  men  earn  $  234  in  5  days,  how  many  men  will 
earn  $  65  in  the  same  time  ? 


WRITTEN  EXERCISES.  225 

9.  If  the  charges  for  transporting  19  cwt.  of  boxed  goods 
from  Harrisburg  to  Philadelphia  are  $  7.03,  what  will  it  cost 
to  transport  36  cwt.  ? 

10.  If  26  acres  of  land  cost  $  1200,  how  much  will  140 
acres  cost? 

11.  If  26  acres  of  land  are  sold  for  $1200,  how  many 
acres  can  be  bought  for  $  5000  at  the  same  rate  ? 

12.  If  it  costs  $120  to  transport  1£  tons  of  freight  590 
miles,  what  will  it  cost  to  transport  2-J-  tons  400  miles  ? 

13.  If  25  bushels  of  oats  last  16  sheep  13  weeks,  how 
long  will  twice  as  many  bushels  last  half  as  many  sheep  ? 

14.  How  many  kegs  of  nails,  @  $  3£  a  keg,  can  I  get  for 
28  barrels  of  sweet  potatoes  @  $  1.75  per  barrel  ? 

15.  What  is  f  of  an  acre  of  land  worth,  if  J  of  an  acre 
is  worth  $80? 

16.  If  T5T  of  a  vessel  is  worth  $  17,000,  how  much  is  £  of 
it  worth  ?     How  much  is  the  whole  vessel  worth  ? 

17.  A  farmer  paid  $50.36  for  building  38f  rods  of  stone 
wall.     How  much  will  it  cost  him  to  build  75  rods  ? 

18.  How  many  pounds  of  butter,  at  24  cents  a  pound, 
must  be  given  in  exchange  for  186  yards  of  muslin  which 
is  sold  at  the  rate  of  15  yards  for  a  dollar  ? 

19.  A  farmer  bought  65  sheep,  at  $  3^  apiece,  and  paid 
for  them  in  hay,  at  $  9£  per  ton.     How  many  tons  of  hay 
was  he  obliged  to  give  in  exchange  for  the  sheep  ? 

20.  How  many  acres  of  land,  at  $45  per  acre,  can  I  get 
for  96  oxen,  at  an  average  price  of  $  42-J-  per  head  ? 

21.  E  and  F  together  own  26,750  acres  of  land.     How 
many  acres  does  each  own,  if  12  times  E's  portion  equals 
13  times  Fs  ? 

22.  A,  B,  and  C  own  97,937  pounds  of  cotton.     How 
much  has  each,  if  A  owns  7  times  as  much  as  C,  and  B  9 
times  as  much  as  C  ? 

STAND.   AR.  — 15 


226  GENERAL  REVIEW  EXERCISES. 

23.  The  sum  of  two  numbers  is  9232,  and  their  difference 
is  427.     What  are  the  numbers  ? 

24.  The  sum  of  two  fractions  is  -J^J,  and  their  difference 
is  Jf .     What  are  the  two  fractions  ? 

25.  A  man  has  two  farms  worth  $  20,491.    The  first  farm 
is  worth  |-  as  much  as  the  second,  plus  $  1560.    What  is  the 
value  of  each  farm  ? 

26.  Seven  times  John's  property,  plus  $32,200,  equals 
21  times  his  property.     How  much  is  he  worth  ? 

27.  If  twice  a  number  is  increased  by  -£%  of  the  number, 
and  2691  more,  the  sum  will  be  three  times  the  number. 
What  is  the  number  ? 

28.  If  5  times  a  number  is  diminished  by  -f-  of  the  num- 
ber, and  1265  more,  the  result  will  be  4  times  the  number. 
What  is  the  number  ? 

29.  What  will  23  acres  8  square  chains  of  land  cost  at 
$37.50  per  acre? 

30.  A  farm  'in  the  form  of  a  trapezoid  contained  100 
acres,  and  its  parallel  sides  were  respectively  80  rd.  and 
120  rd.    What  was  the  distance  between  the  parallel  sides  ? 

31.  How  many  barrels  of  water  will  a  cylindrical  cistern 
hold,  whose  diameter  on  the  bottom  is  6  ft.,  and  whose 
height  is  6  ft.  ? 

32.  A  pile  of  wood  63  ft.  long,  4  ft.  wide,  and  8  ft.  high 
was  sold  at  $  4.75  per  cord.     For  how  much  was  it  sold  ? 

33.  How  much  will  it  cost  to  excavate  a  cellar  35  ft. 
long,  28  ft.  wide,  and  6  ft.  deep  at  $  .45  per  cubic  yard  ? 

34.  What  is  the  cost  of  2  beams  of  timber  each  30  ft. 
long,  10  in.  wide,  and  10  in.  thick  at  $  30  per  M  ? 

35.  A  cubic  foot  of  water  weighs  about  62  Ib.  8  oz.     At 
that  rate  how  much  does  a  barrel  of  water  weigh  ? 


WRITTEN  EXERCISES.  227 

36.  A  and  B  bought  a  horse  for  $  100,  of  which  A  paid 
$  30  and  B  $70.     They  sold  it'  so  as  to  gain  $40.     What 
was  each  one's  share  of  the  gain  ? 

37.  M  and  N  engaged  in  business.     M  furnished  $  900, 
and  N  $  700.     If  they  gained  $  320,  what  was  each  one's 
share  of  the  gain  ? 

38.  C  can  dig  a  well  in  25  days,  and  C  and  D  in  15  days. 
How  long  will  it  take  D  to  dig  what  remains  after  C  has 
dug  £  of  it  ? 

39.  Two  men   enter  into   partnership.      One   furnishes 
$2500  capital,  the  other  $2000.     They  gain  $900.     What 
is  each  one's  share  of  the  gain  ? 

40.  H  and  K  engage  in  trade.     H  furnishes  ^  of  the 
capital,  and  K  the  remainder.     Divide  their  loss  of  $  493 
fairly  between  them. 

41.  A  man  receives  $3  a  day  for  his  labor,  and  pays 
$  .50  a  day  for  his  board.     At  the  expiration  of  30  days  he 
receives  $  60.     How  many  days  was  he  idle  ? 

42.  Mr.  Samuel  Grand  purchased  from  Messrs.  Cox  & 
Davis  24  pr.  kip  boots  @  $  2.50  a  pair,  18  pr.  kid  slippers 
@  $  1.80,  36  pr.  boys'  shoes  @  $  1.20,  and  24  pr.  overshoes 
@  $  .45.     Make  out  a  receipted  bill. 

43.  Two  men  hire  a  pasture  for  $75.     One  pastures  26 
sheep  for  9  weeks,  the  other  37  sheep  for  11  weeks.     What 
should  each  pay  ? 

44.  Three  men  perform  a  piece  of  work.    The  first  works 
36  days ;  the  second,  41  days ;    the  third,  45  days.     They 
receive  $219.60.    How  much  does  each  get,  and  what  are 
the  daily  wages  per  man  ? 

45.  Two  men  engaged  in  the  clothing  business  with  a 
joint  capital  of  $  5000.     The  first  year's  gain  was  $  1760,  of 
which  one   received   $1056.     How   much  capital   did  he 
furnish  ? 


228  GENERAL  REVIEW  EXERCISES. 

46.  What  will  it  cost  to  build  a  wall  42  feet  long,  15  ft. 
high,  and  16  in.  thick,  of  common  bricks  at  $  11.50  per  M, 
laid  in  the  wall,  if  an  allowance  is  made  for  a  gate  10  ft.  by 
10  ft.,  and  the  scaffold  costs  $  9.45? 

47.  There  is  a  wire  fence  inclosing  a  circular  field  80  rods 
in  diameter.     What  will  be  the  area  of  a  square  field  which 
the  same  fence  will  exactly  inclose  ? 

48.  A,  B,  and  C  rent  a  farm  for  $300,  of  which  A  pays 
$80,  B  $100,  and  C  $120.     They  raise   560  bushels  of 
wheat.     What  is  each  man's  share  of  the  wheat  ? 

49.  A  man  has  $3750,  and  owes  $5275.     How  much 
can  he  pay  on  the  dollar,  and  what  will  a  man  get  to  whom 
he  owes  $480? 

50.  Fifteen  persons  agree  to  purchase  a  tract  of  land, 
but  3  of  the  company  withdrawing,  the  investment  of  each 
is  increased  $  150.     What  does  the  land  cost  ? 

51.  A,  B,  and  C  together  invest  $  1200  in  mining  stocks. 
A  puts  in  $280,  B  $365,  and  C  the  remainder.     They  gain 
$  1800.     What  is  the  share  of  each  ? 

52.  The  difference  between  two  numbers  is  17,  and  f  of 
the  first  equals  -J  of  the  second.     What  are  the  numbers  ? 

53.  I  paid  $  650  for  3  horses,  6  cows,  and  20  sheep.    Each 
horse  cost  three  times  as  much  as  each  cow,  and  each  cow 
three  times  as  much  as  one  sheep.     How  much  did  I  pay 
for  each  ? 

54.  A,  B,  and  C  gained  $  700,  of  which  A  received  ^,  B 
^,  and  C  the  remainder.     If  the  whole  capital  was  12  times 
C's  gain,  what  was  the  capital  of  each  ? 

55.  A  man  lost  $600  of  his  money,  and  then  gained  f  as 
much  as  he  had  left.     He  then  had  J  as  much  as  he  had  at 
first.     How  much  had  he  at  first  ? 

56.  A,  B,  and  C  formed  a  partnership.     A   furnished 


WRITTEN  EXERCISES.  229 

$1850,  B  $1950,  and  C  $2050.     They  lost  $2000.     What 
was  each  man's  share  of  the  loss  ? 

57.  A  grocer  in  selling  molasses  uses  as  a  gallon  measure 
one  which  lacks  £  pint  of  being  full  measure.     If  he  sells 
molasses  by  this  measure  to  the.  value  of  $  350,  of  how  much 
money  has  he  cheated  his  customers  ? 

58.  Divide  the  number  12T3^-  into  two  such  parts  that  the 
first  shall  equal  the  second,  plus  2^-. 

59.  A  man  bought  a  number  of  sheep  for  $225;  10  of 
them  having  died,  he  sold  |-  of  the  remainder  for  cost,  and 
received  $  150  for  them.     How  many  did  he  buy  ? 

60.  A  bankrupt  owes  one  of  his  creditors  $  750,  another 
$  820,  and  a  third  $  900.     His  property  amounts  to  $  1500. 
How  much  can  he  pay  on  the  dollar,  and  how  much  will 
each  of  the  creditors  receive  ? 

61.  Two  men  together  receive  $  600  for  grading.     The 
first  furnishes  3  teams  for  15  days,  and  the  second  4  teams 
for  18  days.     How  much  should  each  man  receive  ? 

62.  I  bought  three  farms  for  $15,000.     The  first  cost 
$  400  more  than  the  second,  and  the  third  $  500  less  thaa 
the  second.     What  was  the  cost  of  each  ? 

63.  What  will  be  the  cost  of  the  lumber  to  cover  a  gable 
roof  28  ft.  by  42  ft.  at  $  18  per  M,  if  the  lumber  is  1  inch 
thick  ?     How  many  slates  will  be  required  to  cover  the  roof 
if  3  slates  cover  a  square  foot  ? 

64.  A    man    died    insolvent,   owing    $30,565,   and   his 
property  was  sold  at  auction  for  $  25,000.     How  much  did 
his  estate  pay  on  the  dollar  ? 

65.  D,  E,  and  E  earned  $  3936.     E  earned  three  times  as 
much  as  F,  and  D  four  times  as  much  as  E.     How  much, 
did  each  earn  ? 

66.  How  much  cheaper  will  it  be  to  pave  a  street  \  of  a» 
mile  long  and  60  ft.  wide  with  asphalt  at  $  .22  per  sq.  ft., 
than  to  pave  it  with  granite  blocks  at  $  3.10  per  sq.  yd.  ? 


280  GENERAL   REVIEW   EXERCISES. 

67.  Three  men  contract  to  draw  4815  cords  of  wood  for 
50  cents  a  cord.     If  one  furnished  3  teams,  another  4,  and 
another  5,  what  should  each  man  receive  ?     If  it  took  them 
36  days,  how  much  did  each  team  earn  per  day  ? 

68.  How  many  panels  of  fence  12  ft.  long  will  it  take 
to  inclose  a  rectangular  field  45  rd.  long  and  36  rd.  wide  ? 

69.  Four  persons  rent  a  farm  of  240  A.  96  sq.  rd.  at  $  6£ 
an  acre.     The  first  puts  in  275  sheep,  the  second  320,  the 
third  400,  and  the  fourth  495.   What  rent  should  each  pay  ? 

70.  What  is  the  distance  around  a  water  wheel  if  an  arc 
of  18°  of  its  circumference  is  1  ft.  9  in.  in  length  ?     What 
is  its  diameter  ? 

71.  If  9  men  can  plow  54  acres  in  6  days,  how  many  men 
can  plow  60  acres  in  5  days  ? 

72.  To  find  the  height  of  a  tree,  I  erected  a  stick  3  ft. 
high,  which  cast  a  shadow  1  ft.  9.5  in.     The  shadow  of  the 
tree  at  the  same  time  was  48  ft.  10  in.    What  was  its  height  ? 

73.  How  many  square  feet  of  boards  are  required  for 
the  two  gables  of  a  barn  32  ft.  wide,  the  ridge  being  8  ft. 
6  in.  higher  than  the  plates  ? 

74.  How  much  will  it  cost  to  build  two  abutments  for  a 
bridge,  each  18  ft.  long,  12  ft.  wide  at  the  bottom,  8  ft. 
wide  at  the  top,  and  11  ft.  high,  at  $2.50  a  perch? 

75.  C  and  D  have  the  same  income.     C  spends  ^  of  his, 
but  D,  by  spending  $65  more  each  year  than  C,  at  the  end 
of  a  year,  finds  himself  $  10  in  debt     How  much  does  each 
spend  yearly  ? 

76.  A    man    willed    his    estate    worth    $8000    as    fol- 
lows:  to  his  wife  £,  to  his  daughter-^,    and   to    his  son 
•jSg-.     If  the  widow  should   die,   leaving  her   share  to  son 
and  daughter   in  the   same   proportion,  how  much  in  all 
would  each  receive? 


PERCENTAGE. 


273.  1.  A  merchant  lost  $4  out  of  every  $100  worth  of 
goods  sold  on  account  of  bad  debts.  What  part  of  his  sales 
did  he  lose  ? 

2.  A  man  spent  $5  out  of  every  $10  that  he  earned. 
How  many  hundredths  of  his  earnings  did  he  spend  ? 

3.  A  company  of  soldiers  engaged  in  battle  lost  1  out  of 
every  10  men.     How  many  was  that  per  hundred  or  per  cent  f 

4.  In  a  school  6  out  of  every  10  pupils  are  girls.     How 
many  hundredths  of  the  number  are  girls  ?     What  percent  ? 


5.    What  is  y^,  or  5  per  cent,  of  $500?    Of  $1000?    Of 
$2000?     Of  $3000?     Of  $5000? 


6.  What  is  yfc,  or  2  per  cent,  of  $  400  ?     Of  $  600  ?    Of 
$1000?     Of  $1200?     Of  $1600? 

7.  What  is  6  per  cent  of  $  600  ?     Of  $  800  ?    Of  $1000  ? 

274.  The  expression  Per  Cent  means  by  the  hundred. 

It  is  a  contraction  of  the  Latin  per  centum,  by  the  hundred. 

275.  The  commercial  Sign  of  Per  Cent  is  %. 
Thus,  8  %  is  read  8  per  cent. 

276.  That  part  of  arithmetic  which  treats  of  processes 
involving  per  cent  is  termed  Percentage. 

231 


232  PERCENTAGE. 

277,   Since  per  cent  is  a  number  of  hundredths,  it  is 
usually  expressed  as  a  decimal. 

It  may  also  be  expressed  as  a  common  fraction.     Thus, 

5  per  cent  is  written      5%,    .05, 
10  per  cent  is  written    10%,    .10, 

per  cent  is  written  12£%,  .12£,  .125, 


1UU 

f  per  cent  is  written  f  %,  .OOf  ,  .006,          or 
215  per  cent  is  written  215%,  2.15. 

278.    Express  decimally  : 


1.  10%. 

7.     6J%. 

13.  125%. 

19.     |%. 

2.  15%. 

8.     81%. 

14.  132%. 

20.     f%. 

3.  20%. 

9.  12|%. 

15.  148%. 

21.  A%- 

4.  25%. 

10.  181%. 

16.  210%. 

22.  -fa%. 

5.  50%. 

11.  17|%. 

17.  275%. 

23.     |%. 

€.  75%. 

12.  25|%. 

18.  287]-%. 

24.  fi%. 

279.   Express  by  common 

fractions  in  their 

lowest  terms  ; 

1.  20% 

7.  16f%. 

13.  150%. 

19.     |%. 

2.  35%. 

8.  33£%. 

14.  165%. 

20.     |%. 

3.  42%. 

9.  37^%. 

15.  175%. 

21.  A%. 

4.  56%. 

10.  62|%. 

16.  180%. 

22.  ft%. 

5.  75%. 

11.  83£%. 

17.  225%. 

23.  ££%. 

6.  85%. 

12.  87£%. 

18.  375%. 

24.  «#. 

DEFINITIONS.  233 

280.  Express  in  per  cent  decimally : 

1.  i.  5.  J.  9.  f  13.  £.  17.  f. 

2.  i.  6.  f  10.  f.  14.   A.  18.  TV 

3.  f  7.  1  11.  ^.  15.  TV  19.  ^. 

4.  ^  8.  f.  12.  f  16.  f  20.  ^. 

Problems  in  percentage  involve  the  following  elements : 

281.  The  number  of  which  the  per  cent  is  to  be  found,  or 
the  Base. 

282.  The  number  of  hundredths  taken,  or  the  Rate. 

283.  The  number  which  is  a  certain  number  of  hun- 
dredths of  the  base,  or  the  Percentage. 

284.  The  sum  of  the  base  and  percentage,  or  the  Amount. 

285.  The  base  less  the  percentage,  or  the  Difference. 

In  the  formulas,  B  represents  the  base  ;  E,  the  rate ;  P,  the  per- 
centage ;  A,  the  amount ;  and  D,  the  difference. 

286.  To  find  the  percentage  when  the  base  and  rate  are 
given. 

Find: 

1.  10%  or  T%  or  ^  of  $  150. 

2.  25%  or^or  *  of  $320. 

3.  50%  or  ^  or  \  of  $600. 

4.  12^%  or  i?i  or  |  of  $160. 

100 

Find: 

5.  40%  of  250  tons.  7.   331%  of  660  Ib. 

6.  15%  of  300  mi.  8.    60%  of  200  rd. 


234  PERCENTAGE. 

9.  20%  of  200  bu.  13.  65%  of  500  bricks. 

10.  30%  of  400  gal.  14.  75%  of  600  sheep. 

11.  50%  of  400  A.  15.  80%  of  120  horses. 

12.  25%  of  800  yd.  16.  90%  of  360  days. 

WRITTEN   EXERCISES. 

1.  What  is  12J%  of  $864.80  ? 

EXPLANATION.  —  Since  12 £%  of  a 
number  is  .12|  of  it,  12£%  of  $864.80 
$ 864.80  x. 121= $108.10,    is  -12£  of  $864.80,  or  $108.10. 

°r'  121 

|   of   $  864.80*=  $  108.10.       Since  12£  %  of  a  number  **  ^  or  i 

of  it,  12 1  %  of  $864.80  is  \  of  $864.80, 
which  is  $  108.10. 

FORMULA,  BxR  =  P. 

Find: 

2.  15%  of  $975.40.'  5.  37^%  of  $1260. 

3.  50%  of  $858.50.  6.  125%  of  $1864. 

4.  1%  of  $576.40.  7.  If  %  of  $2520. 

8.  A  man  bought  350  cows,  and  then  sold  60%  of  them. 
How  many  cows  had  he  left  ? 

9.  A  farmer  had  375  acres  of  land,  and  sold  3£|%  of 
it.     How  much  remained  ? 

10.  B  bought  3480  barrels  of  flour,  and  then  sold  16f  % 
of  it.     How  many  barrels  remained  ? 

11.  C,  having  a  flock  of  575  sheep,  sold  64%  of  them. 
How  many  had  he  left  ? 

12.  A  man's  income  is  $  900  a  year,  and  he  spends  67£% 
of  it.     How  much  does  he  spend  ? 


WRITTEN   EXERCISES.  235 

13.  D  bought  1320  acres  of  land,  and  sold  37%%  of  it. 
How  much  had  he  left  ? 

14.  In  a  school  of  400  pupils,  62^%  are  boys.     How 
many  boys  are  there  in  the  school  ? 

15.  How  much  metal  will  be  obtained  from  375  tons  of 
ore  if  the  metal  is  10^%  of  the  ore  ? 

16.  A  farm  cost  $4750,  but  since  it  was  purchased  the 
farm  has  decreased  16^-%  in  value.     What  is  the  present 
value  of  the  farm  ? 

17.  If  cloth  will  shrink  5^-%  of  its  length  in  sponging, 
what  will  be  the  shrinkage  of  a  piece  of  cloth  containing  42 
yards  before  sponging  ? 

18.  If  a  man's  salary  is  $1350  a  year,  and  his  expenses 
are  87|-%  of  that  sum,  how  much  can  he  save  yearly  ? 

19.  A  man  having  $27,000,  invested  18%  in  bank  stock, 
12^-%  in  bonds  and  mortgages,  34%  in  town  property,  and 
the  rest  in  a  farm.     How  much  did  the  farm  cost  ? 

20.  Mr.  B  receives  a  salary  of  $  1800  a  year.     He  pays 
15%  of  it  for  board,  8-|%  for  clothing,  and  16%  for  other 
•expenses.     What  are  his  yearly  expenses,  and  how  much 
does  he  save  ? 

21.  A  man  bequeathed  15%  of    his   estate  to  a  col- 
lege, 10%  to  an  asylum,  10%  to  a  church,  5%  to  a  public 
library,  and  the  remainder  he  divided  equally  among  his  7 
children.     What  did  each  child  receive,  the  estate  being 
worth  $150,000? 


L$  150,000? 

L  To  find  the  rate  when  the  base  and  percentage  are  given. 

1.  What  part  of  25  is  5  ?     How  many  hundredths  of  25 
is  5  ?    What  per  cent  ? 

2.  What  part  of  40  is  4?     How  many  hundredths? 
What  per  cent  ? 


287 

l. 


236  PERCENTAGE. 

3.  What  part  of  60  is  15  ?     How  many  per  cent  ? 

4.  What  per  cent  of  35  is  7  ?     Of  50  is  10  ? 
What  per  cent  of 

5.  24  is  12?       11.   $50  are  $30?  17.  fisf? 

6.  36  is  12  ?       12.   60  qt.  are  15  qt.  ?         18.  f  is  f  ? 

7.  40  is  8  ?         13.    75  gal.  are  25  gal.  ?     19.  ft  is  ^-  ? 

8.  50  is  20  ?       14.    64  bu.  are  16  bu.  ?       20.  ^  is  ^  ? 

9.  80  is  40  ?       15.   72  Ib.  are  36  Ib.  ?        21.  ^  is  ^5T  ? 
10.  90  is  30  ?       16.   96  ft.  are  32  ft.  ?         22.  £  is  £  ? 

23.  A  farmer  who  had  40  sheep,  sold  15.     What  per  cent 
did  he  sell  ?     What  per  cent  remained  unsold  ? 

24.  From  a  farm  of  150  acres,  30  acres  were  sold.     What 
per  cent  of  it  was  sold  ?     What  per  cent  remained  unsold  ? 

WRITTEN   EXERCISES. 

1.   A  merchant  who  had  450  yd.  of  muslin  sold  150  yd. 
What  per  cent  of  it  did  he  sell  ? 

SOLUTION.  — 150  yd.  =  £f  §,  or  i,  or  331  %  of  450  yd. 

Or, 

1%  of  450  yd.  =  4.5  yd. 

/.  150  yd.  is  as  many  per  cent  of  450  yd.  as  4.5  yd.  is  contained  times 
in  150  yd.,  or33i%. 

FORMULA,  P  -f-  B  =  It. 

What  per  cent  of 

2.  840  men  are  420  men  ?  8.  680  rd.  are  510  rd.  ? 

3.  450  hr.  are  150  hr.  ?  9.  876  gal.  are  584  gal.  ? 

4.  560  min.  are  140  min.  ?  10.  f  is  ^  ? 

5.  720  bu.  are  120  bu.  ?  11.  f  is  ^  ? 

6.  540  A.  are  210  A.  ?  12.  ^  is  \  ? 

7.  450  T.  are  300  T.  ?  13.  ^  is  £  ? 

14.   A  sheep  grower  sold  75  sheep  from  a  flock  of  300. 
What  per  cent  of  the  flock  did  he  sell  ? 


WRITTEN  EXERCISES.  237 

15.  Out  of  350  words  a  student  spelled  329  correctly. 
What  %  of  the  words  were  spelled  correctly  ? 

16.  A  fruit  grower  transplanted  275  peach  trees,  and  40 
of  them  died.     What  %  of  them  died  ? 

17.  A  farmer  raised  250  bushels  of  oats,  and  sold  all  but 
65  bushels.     What  %  of  his  crop  did  he  sell  ? 

18.  In  a  journey  of  1560  miles,  Mr.  D  traveled  195  miles 
by  stage,  and  the  rest  of  the  distance  by  rail.     What  % 
of  the  distance  did  he  travel  by  rail  ? 

19.  A  grocer  having  on  hand  1200  pounds  of  sugar,  sold 
•J-  of  it  at  one  time,  and  £  of  the  remainder  at  another  time. 
What  %  of  the  whole  remained  unsold  ? 

20.  A  farmer  raised  560  bushels  of  grain.     He  sold  A 140 
bushels ;  \  of  the  remainder  he  sold  to  B,  and  f  of  what 
still  remained  to  C.     What  %  of  the  whole  had  he  left  ? 

21.  A  farmer  raised  120  bushels  of  potatoes  from  4  bush- 
els of  seed.     What  %  of  the  crop  was  the  seed  ? 

22.  A  man  paid  $  15  for  the  use  of  land  which  cost  $  275. 
What  %  does  the  land  owner  realize  on  his  investment  ? 

23.  A  gentleman,  finding* that  he  was  constantly  growing 
deeper  in  debt,  to  save  his  creditors,  made  an  assignment. 
His  property  was  worth  $  5765.85,  and  his  debts  amounted 
to  $  7125.     What  %  was  paid  to  the  creditors  ? 

24.  A  man's  salary  is  f  4000.     He  spends  22%  for  fuel 
and  rent,  12%  for  clothing,  3%  for  books,  and  $1018  for 
other  purposes.     What  %  of  his  salary  has  he  left  ? 

288.   To  find  the  base  when  the  percentage  and  rate  are 
given. 

1.  Of  what  sum  is  $  10  25%  or  ^  or  £  ? 

2.  Of  what  sum  is  40  20%  or  ^  or  -J-  ? 


288  PERCENTAGE. 

3.  Of  what  number  is  80  40%  or  ^  or  |  ? 

4.  Of  what  number  is  30  12|%  or  -J-  ? 
Find  the  number  of  which  : 

5.  20  is  20%.  -      11.     50  is    25%.     17.  f  is    50%. 

6.  45  is    5%.         12.     60  is    60%.      18.  f  is    25%. 

7.  60  is    1%.         13.     60  is    75%.      19.  f  is 

8.  25  is    |%.         14.     30isl6f%.      20.  ^  is 

9.  40  is  10%.         15.     80is66f%.     21.  .5  is    10%. 
10.  40  is  40%.         16.    210  is  871%.      22.  .05  is      £%. 

23.  My  expenses  were  $400,  which  sum  was  62£%  of 
my  income.     How  much  was  my  income  ? 

24.  A  fire  destroyed  33|%  of  a  stock  of  goods,  and  the 
value  of  the  goods  destroyed  was  $2000.     What  was  the 
value  of  the  entire  stock  ? 

25.  The  attendance  at  a  school  was  35  less  than  the  whole 
number  of  pupils  enrolled,  and  the  absentees  were  25%  of 
the  enrollment.     How  many  pupils  attended  the  school  ? 

WRITTEN    EXERCISES. 

1.  A  magazine  contained  3B  pages  of  advertisements, 
which  was  30%  of  the  whole  number  of  pages.  How  many 
pages  were  there  in  the  magazine  ? 

SOLUTION. 

30%  or  T%%  or  T^  of  the  number  of  pages  =  36. 
/.  ^  of  the  number  of  pages  =  12. 
And  the  whole  number  of  pages  =  120. 

Or, 

30  %  of  the  number  of  pages  =  36. 
.*.  1  %  of  the  number  of  pages  =  ^  of  36,  or  1.2. 
And  the  whole  number  of  pages  =  100  times  1.2,  or  120. 
FORMULA,  P  -r-  R  =  B. 


WRITTEN   EXERCISES.  23$ 

Find  the  number  of  which : 

2.  102  is  24%.  9.  7.5  is  f  %.  16.  86.5  is  16f  %. 

3.  228  is  9£%.  10.  11.8  is  £%.  17.  45  is  1.5%. 

4.  8.25  is  2%%.  11.  817  is  19%.  18.  f  is  1£%. 

5.  55.9  is  13%.  12.  .96  is  |%.  19.  *  ia  7.5%. 

6.  798  is  33J%  13.  2.17  is  31%.  20.  36  is  £%. 

7.  896  is  112%.    14.  812  is  175%.  21.  44  is  .4%. 

8.  981  is  90%.  15.  1250  is  200%.  22.  7  is  2J%. 

23.  A  farmer  sold  786  bushels  of  potatoes,  which  were 
75%  of  his  entire  crop.     How  many  bushels  did  he  raise  ? 

24.  A  merchant  sold  25%  of  his  goods  for  $6650.     At 
this  rate  what  were  the  goods  worth  ? 

25.  If  a  man  rents  a  house  for  $520  per  annum,  which 
is  13%  of  the  value  of  the  property,  what  is  its  value  ? 

26.  A  farmer  sold  315  bushels  of  wheat,  which  was  30% 
of  his  crop.     What  was  his  entire  crop  ? 

27.  A  man  spends  $1239  a  year,  which  is  84%  of  his 
salary.     What  is  his  salary  ? 

28.  A  teacher  spends  65%  of  his  income,  and  can  thus 
save  $  420.     What  is  his  income  ? 

29.  Mr.  Kook  sold  a  horse  for  $225,  which  was  90%  of 
what  he  paid  for  it.     What  did  the  horse  cost  him  ? 

30.  A  merchant-vessel  has  289  tons  of  leather  on  board, 
which  is  17%  of  the  whole  cargo.     How  many  tons  of  cargo 
does  she  carry  ? 

31.  The  distance  between  two  stations  on  a  certain  rail- 
road is  14.5  miles,  which  is  12^%  of  the  whole  length  of 
the  road.     What  is  the  length  of  the^road  ? 


240  PERCENTAGE. 

32.  An  assignee  paid  off  debts  to  the  amount  of  $870, 
which  was  33£%  of  the  total  indebtedness.     What  was  the 
total  indebtedness  ? 

33.  Into   a  vessel  containing  pure  vinegar  there  were 
thrown  12  J  gallons  of  water,  which  was  18f  %  of  the  dilu- 
tion.    What  was  the  quantity  of  pure  vinegar  ? 

34.  On  Monday  a  merchant's  sales  amounted  to  $  185.70, 
and  that  sum  was  12|  %  of  his  sales  for  the  week.     How 
much  were  his  sales  for  the  week  ? 

35.  A  farmer  sold  1240  bushels  of  corn,  which  was  62^-% 
of  the  number  of  bushels  raised.     How  much  did  he  raise  ? 

289.   To  find  the  base  when  the  amount  and  rate  are  given. 

1.  If  a  merchant  gains  20%  of  the  cost  in  selling  goods, 
how  many  per  cent  of  the  cost  does  he  get  for  them  ? 

2.  If  a  sum  of  money  is  increased  by  25%,  or  -^  of  itself, 
how  many  per  cent  of  the  original  sum  will  it  be  ?    What 
part? 

3.  A  boy  who  bought  20%  as  many  marbles  as  he  had, 
found  that  he  then  had  60.     How  many  had  he  at  first  ? 

4.  A  clerk's  wages  were  increased  10%,  and  he  then 
received  $66  per  month.     What  were  his  wages  before  the 
increase  ? 

What  number  increased  by : 

5.  10%  of  itself  =  55  ?  11.  12£%  of  itself  =  90  ? 

6.  30%  of  itself  =  26  ?  12.  37£%  of  itself  =  77  ? 

7.  50%  of  itself  =  48  ?  13.  16f  %  of  itself  =  56  ? 

8.  20%  of  itself  ==  60  ?  14.  25%  of  itself  =  2£  ? 

9.  25%  of  itself  =  50  ?  15.  40%  of  itself  =  2^  ? 
10.  33£%  of  itself  =  60  ?  16.  60%  of  itself  =   f  ? 


WRITTEN   EXERCISES.  241 

17.  A  bookseller  sold  a  book  for  $1.25,  gaining  25%  of 
the  cost.     What  did  it  cost  him  ? 

18.  The  retail  price  of  molasses  is  50  cents  per  gallon, 
and  the  merchant  makes  a  profit  upon  it  of  25%  of  the  cost. 
What  is  the  cost  ? 

WRITTEN    EXERCISES. 

1.  What  number  increased  by  27%  of  itself  equals  508  ? 
SOLUTION.  — Since  the  number  is  increased  by  27%  of  itself, 

127%  of  the  number  =  508. 
.*.  1  %  of  the  number  =  4. 
And  the  number  =  400. 

Or, 

f|£  of  the  number  =  508. 
•*•  Ttffl  °^  ^ne  number  =  4. 
And  the  number  =  400. 

FORMULA,  B  =  A  -j-  (1  +  R). 

What  number  increased  by : 

2.  12%  of  itself  =  560?  7.  16f%  of  itself  =    700? 

3.  17%  of  itself  =  702?  8.  2J%  of  itself  =    820? 

4.  31%  of  itself  =  786  ?  9.  20%  of  itself  =  1620  ? 

5.  38%  of  itself  =  966?  10.  37-i-%  of  itself  =    682? 

6.  62%  of  itself  =  648?  11.  100%  of  itself  =  1796  ? 

12.  A  has  372  sheep,  which  are  20%  more  than  B  has. 
How  many  sheep  has  B  ? 

13.  A  clerk's  salary  was  increased  15%,  and  now  it  is 
$  1050.     What  was  his  former  salary  ? 

14.  A  farmer  raised  750  bushels  of  corn,  which  was  50% 
more  than  the  number  of  bushels  of  wheat  raised.     How 
many  bushels  of  wheat  did  he  raise  ? 

STAND.    AR.  —  16 


242  PERCENTAGE. 

15.  A  grocer  expended  $  36.48  for  vegetables,  which  was 
33%%  more  than  he  expended  for  butter  and  eggs.     How 
much  did  he  expend  for  butter  and  eggs  ? 

16.  A  drover  sold  cows  and  sheep  for  $6105.      If  he 
received  65%  more  for  the  cows  than  for  the  sheep,  how 
much  did  he  get  for  the  cows  ? 

17.  A  house  cost  $2072,  which  is  40%  more  than  a  barn 
cost.     What  was  the  cost  of  the  barn  ? 

18.  B  bought  a  farm  for  a  certain  sum.     He  expended 
for  stock  11%  of  the  price  of  the  farm,  and  found  that  the 
cost  of  both  was  $  8214.     What  did  he  pay  for  the  farm  ? 

19.  C  raised  496  bushels  of  wheat,  which  was  33^%  more 
than  |  of  what  D  raised.     How  many  bushels  did  D  raise  ? 

20.  The  population  of  a  certain  town  is  8118,  which  is 
121%  more  than  it  was  three  years  ago.     What  was  the 
population  then  ? 

21.  A  drover  sold  275  sheep  for  $  1375,  which  was  37|% 
more  than  they  cost.     What  did  the  sheep  cost  per  head  ? 

22.  E  bought  a  farm  for  $  7402.50,  which  was  17^%  more 
than  F  paid  for  his  farm.     How  much  did  F  pay  for  his 
farm? 

290.    To  find  the  base  when  the  difference  and  rate  are  given. 

1.  If  a  merchant  loses  20%  of  the  cost  in  selling  goods, 
what  per  cent  of  the  cost  does  he  get  for  them  ? 

2.  If  a  sum  of  money  is  decreased  by  25%,  or  \  of 
itself,  how  many  per  cent  of  the  original  sum  is  it  ?     What 
part? 

3.  A  lad  lost  20%,  or  £  of  his  marbles,  and  found  that 
he  had  40  marbles  left.     How  many  had  he  at  first  ? 


WRITTEN  EXERCISES.  243 

4.  A  clerk  whose  wages  had  been  reduced  10%  was 
receiving  $  63  per  month.    What  were  his  wages  before  the 
reduction  ? 

What  number  decreased  by 

5.  10%  of  itself  =  90?          11.  30%  of  itself  =  56? 

6.  20%  of  itself  =  48  ?          12.  25%  of  itself  =  60  ? 

7.  33i%  of  itself  =  80  ?        13.  50%  of  itself  =  72  ? 

8.  12J%  of  itself  =  70  ?       14.  50%  of  itself  =  7*  ? 

9.  37|%  of  itself  =  65?        15.   40 %  of  itself  =  4J-  ? 
10.   16|%  of  itself  =75?        16.   25%  of  itself  =  £|? 

17.  A  merchant  who  sold  his  goods  at  20%  below  cost, 
received  80  cents  per  yard  for  silk.     What  did  it  cost  him  ? 

18.  The  retail  price  of  bananas  was  16  cents  per  dozen, 
but  the  price  was  20%  below  cost.     What  did  they  cost? 

WRITTEN   EXERCISES. 

1.  What  number  diminished  by  40  %  of  itself  equals  432  ? 
SOLUTION.  —  Since  the  number  is  diminished  by  40  %  of  itself, 

.-.  60%  of  the  number    =  432. 
1  %  of  the  number  =  7.2. 
And  the  number  =  720. 
Or, 

T%%  or  I  of  tlie  number  =  432. 
£  of  the  number  =  144. 
And  the  number  =  720. 

FORMULA,  B  =  D  -T-  (1  -  K) . 

What  number  diminished  by 

2.  25%  of  itself  =  270  ?  5.  15%  of  itself  =  544 ? 

3.  50%  of  itself  =  325  ?  6.  90%  of  itself  =  12.6  ? 

4.  30%  of  itself  =  427  ?  7.  4£%  of  itself  =  382  f 


244  PERCENTAGE. 

8.  A  boy  spent  25%  of  his  money,  and  then  had  $  12.36 
left.     How  much  had  he  at  first  ? 

9.  A  clerk  after  spending  65%  of  his  salary  had  $385 
left.     What  was  his  salary  ? 

10.  Mr.  Kirk  sold  340  bushels  of  wheat,  and  had  15%  of 
it  left.     How  many  bushels  had  he  at  first  ? 

11.  A  clothier  sold  a  suit  for  $31.20,  which  was  20% 
less  than  the  price  he  asked  for  it.     What  was  his  asking 
price  ? 

12.  A  man  spent  $  45.75,  and  then  had  40%  of  his  money 
left.     How  much  had  he  at  first  ? 

13.  The  expenses  of  a  firm  during  the  month  of  Novem- 
ber were  $185.68.     The  expenses  in  November  were  12% 
less  than  during  the  month  of  December.     What  were  the 
•expenses  in  December  ? 

14.  Mr.  Hallam  deposited  85%  of  his  money  in  a  bank, 
and  afterward  drew  out  20%  of  the  sum  deposited,  and  then 
had  $  3859  in  the  bank.   What  was  the  amount  of  his  money  ? 

15.  35%  of  a  regiment  being  sick,  only  637  men  were 
able  to  enter  battle.     How  many  men  were  there  in  the 
regiment  ? 

16.  At  a  forced  sale  a  bankrupt  sold  his  farm  for  $7500, 
which  was  33 J%  less  than  its  real  value.     What  was  the 
value  of  the  farm  ? 

17.  A  man  bought  1785  locust  posts,  which  was  62^% 
less  than  the  number  of  chestnut  rails  purchased.     How 
many  chestnut  rails  did  he  buy  ? 

18.  A  merchant's  sales  on  Monday  amounted  to  $385.84. 
His  sales  on  Monday  were  16f  %   of  54%   less  than  the 
amount  of  goods  sold  on  Tuesday.    What  was  the  amount 
of  Tuesday's  sales  ? 


PROFIT  AND  LOSS.  245> 

PROFIT  AND  LOSS. 
291.   1.    When  25%  is  gained,  what  part  is  gained  ? 

2.  For  what  must  sugar  that  cost  5  cents  per  Ib.  be  sold 
so  as  to  gain  20%  of  the  cost  ?     What  is  the  gain  per  Ib.  ? 

3.  A  merchant  sold  goods  that  cost  him  25  cents  per 
yd.  at  a  loss  of  10%  of  the  cost.     What  was  the  loss  ? 

4.  If  boots  that  cost  $4  a  pair  are  sold  for  $5,  what 
part  of  the  cost  is  gained  ?     What  per  cent  ? 

5.  When  I  sell  butter  for  25  cents  per  Ib.  that  cost  20 
cents,  what  part  of  the  cost  do  I  gain  ?    What  per  cent  ? 

6.  Grain  that  cost  60  cents  per  bu.  was  damaged  so  that 
it  was  sold  for  40  cents  per  bu.     What  per  cent  of  the  cost 
was  lost  ? 

7.  Dictionaries  that  cost  $6  were  sold  for  $4.     What 
per  cent  of  the  cost  was  lost  ? 

8.  By  selling  goods  at  a  gain  of  5  cents  per  yd.,  25% 
or  ^  of  the  cost  was  gained.    What  was  the  cost  ? 

9.  Goods  were  sold  at  a  loss  of  20%  of  the  cost.     If  the 
loss  per  yd.  was  4  cents,  what  was  the  cost  per  yd.  ? 

10.  A  grain  dealer  sold  wheat  at  a  gain  of  8  cents  per 
bu.,  which  was  an  advance  of  10%  of  the  cost.     What  was 
the  cost  ? 

11.  The  loss  upon  a  quantity  of  damaged  books  was  25 
cents  per  copy,  and  that  sum  was  12|-%  of  the  cost.     What 
did  they  cost  apiece  ? 

12.  A  horse  was  sold  for  $  75  less  than  it  cost.     If  the 
loss  was  25%  of  the  cost,  what  was  the  cost  ? 

13.  Flour  was  sold  at  a  profit  of  12£%  of  the  cost,  and 
the  gain  per  barrel  was  80  cents.    What  did  it  cost  per  bbl.  ? 


246  PERCENTAGE. 

14.  A  grocer  made  a  profit  of  6  cents  per  Ib.  by  selling 
tea  at  an  advance  of  20%  of  the  cost.     What  did  it  cost  ? 

15.  By  selling  flour  at  $  7  per  barrel,  16f%  of  the  cost 
was  gained.     What  did  the  flour  cost  ? 

SOLUTION.  —  Since  16f%,  or  £  of  the  cost  was  gained,  the  selling 
price  must  have  been  |  of  the  cost.  Since  f  of  the  cost  was  $  7,  £  was 
}  of  $7,  or  $  1.  And  since  £  of  the  cost  was  $  1,  the  entire  cost  was  $  6. 


16.  By  selling  silk  at  90  cents  per  yd.,  10%  of  the  cost 
was  lost.     What  did  the  silk  cost  per  yd.  ? 

17.  A  man's  profits  upon  bicycles  was  40%  of  the  cost 
when  he  sold  them  at  $  140  each.     What  did  they  cost  ? 

18.  I  sold  goods  at  $  1.25  per  yd.,  and  gained  25%  of  the 
cost.     What  did  the  goods  cost  ? 

19.  A  tennis  outfit  cost  me  $  24.     If  the  dealer  who  sold 
it  made  a  profit  of  33  J%,  how  much  did  it  cost  him  ? 

20.  A  pair  of  skates  cost  a  boy  $  2.50,  but  the  merchant 
sold  them  at  a  gain  of  25%.     What  did  they  cost  him  ? 

21.  A  coat  was  sold,  at  a  loss  of  16J%  of  the  cost,  for 
$  15.     How  much  did  it  cost  ? 

22.  A  carriage  was  sold  for  .$25  more  than  it  cost.     If 
that  sum  was  25%  of  the  cost,  what  was  the  cost  ? 

23.  A  clothier  sold  a  coat  for  $25  that  cost  him  $20. 
What  per  cent  of  the  cost  did  he  gain  ? 

24.  A  drover  averaged  a  profit  of  $  10  per  head  upon  a 
drove  of  cattle.     If  that  was  25%  of  the  cost,  what  did  they 
cost  him  per  head  ? 

25.  A  merchant  wishes  to  sell  lumber  that  oost  $  15  per  M 
at  a  gain  of  33^%  of  the  cost.     What  must  he  get  for  it  ? 

26.  Furniture  that  cost  $  80  was  sold  at  a  loss  of  12^%. 
For  how  much  was  it  sold  ? 

292.   PRINCIPLE.  —  Gain  or  loss  is  reckoned  at  a  certain 
per  cent  of  the  cost,  or  sum  invested. 


X 

PROFIT  AND   LOSS.  247 

WRITTEN   EXERCISES. 

1.  A  grocer  bought  a  hogshead  of  sugar  for  $  48.93,  and 
sold  it  at  a  gain  of  15%.     What  was  his  gain  ? 

2.  A  man  bought  a  team  for  $  350,  and  sold  it  at  a  gain 
of  20%.     What  did  he  receive  for  it  ? 

3.  A  dealer  invested  $  2460  in  shoes,  and  sold  them  at 
a  gain  of  25%.     How  much  did  he  gain  ? 

4.  A  farm  was  bought  for  $4675,  and  sold  at  a  gain  of 
8%.    What  was  the  gain  ? 

5.  A  piano  that  cost  $215  was  sold  at  a  gain  of  40%. 
For  how  much  was  it  sold  ? 

6.  A  drover  paid  $  1890  for  cattle  that  he  was  obliged 
to  sell  at  a  loss  of  16|%.     What  was  the  loss  ? 

7.  A  house  that  cost  $3600  was  sold  at  a  gain  of  18  %, 
How  much  was  received  for  it  ? 

8.  A  merchant  sold  goods  that  cost  $2180,  at  a  gain  of 
33J%.     How  much  did  he  receive  for  them  ? 

9.  A  ship  that  cost  $  115,000  was  sold  at  a  loss  of 
How  much  was  received  for  it  ? 


10.  I  bought  1260  Ib.  of  sugar  at  4J^  a  pound,  and  sold 
it  at  a  gain  of  10%.     How  much  did  I  sell  it  for  ? 

11.  E  bought  2360  bu.  of  wheat  at  $.85  per  bushel,  and 
sold  it  at  a  gain  of  18f  %.     How  much  did  he  get  for  it  ? 

12.  What  per  cent  is  gained  by  selling  tea  at  70  cents 
per  Ib.  that  cost  60  cents  ? 

SOLUTION.  $.70  -  $.60  =  $.10,  gain. 

$.10-$.60  =  16f%. 

Or, 
The  part  of  the  cost  gained  is  £$,  or  £,  which  is  equal  to  16|%» 


.' 
248  PERCENTAGE. 

13.  Flour  that  cost  $4.50  per  barrel  was  sold  for  $  4.95 
per  barrel.     What  was  the  gain  per  cent  ? 

14.  A  house  that  cost  $  3200  was  sold  for  $  3680.     What 
Tvas  the  gain  per  cent  ? 

15.  A  man  whose  salary  was  $2400  had  it  increased  to 
$  3000.     What  was  the  per  cent  of  increase  ? 

16.  A  dealer  paid  $650  for  200  tons  of  coal,  and  sold  it 
for  $  780.     What  was  the  gain  per  cent  ? 

17.  A  merchant  bought  350  yards  of  silk  at  $1.12J  a 
yard,  and  sold  it  at  a  profit  of  $  131.25.     What  per  cent 
did  he  gain  ? 

18.  A  merchant  bought  goods  at  20%   less  than  their 
market  value,  and  sold  them  at  20%  above  market  value. 
What  was  his  gain  per  cent  ? 

19.  A  horse  that  cost  $145.50  was  sold  for  $194.     What 
was  the  gain  per  cent  ? 

20.  What  is  the  gain  per  cent  if  goods  which  cost  $  7500 
are  sold  for  $  9375  ? 

21.  A  man  paid  $  6450  for  a  farm,  and  spent  on  improve- 
ments a  sum  equal  to  60  per  cent  of  the  purchase  price. 
He  then  sold  the  farm  for  $  11,868.     What  was  the  gain  per 
cent  on  the  whole  cost  ? 

22.  A  horse  was  sold  at  an  advance  of  $  75,  which  was  a 
gain  of  25%.     What  was  the  cost  of  the  horse? 

SOLUTION.  — 25  %  or  .25  or  i  of  the  cost  of  the  horse  =  $  75. 
.-.  The  cost  of  the  horse  =  $  300. 

23.  A  jeweler  made  $20  on  a  watch  by  selling  it  at  a 
profit  of  40%.     What  did  the  watch  cost  ? 

24.  A  painting  was  sold  for  $  3.60  less  than  cost,  which 
was  a  loss  of  20%.     What  was  the  cost  ? 


PROFIT   AND   LOSS.  249 

25.  By  selling  cloth,  at  an  advance  of  28^  per  yard,  I 
make  a  profit  of  25%.     What  did  it  cost  ? 

26.  A  dealer  sold  a  quantity  of  wheat  at  a  profit  of  12^-%, 
and  gained  $  250.     What  was  the  cost  of  the  wheat  ? 

27.  A  grocer  lost  8%  by  selling  56  pounds  of  butter  for 
$  1.12  less  than  cost.     What  did  it  cost  him  per  pound  ? 

28.  A  farmer  gained  15%  by  selling  land  at  an  advance 
of  $  11.25  per  acre.     What  was  the  cost  per  acre  ? 

29.  A  man  lost  16f  %  by  selling  a  house  for  $538  less 
than  cost.     What  did  it  cost  ? 

30.  The  gain  on  a  quantity  of  lumber  was  $  918.75,  which 
was  a  profit  of  21%.    What  was  the  cost  of  the  lumber  ? 

31.  A  speculator  gained  $  1650  by  selling  land  at  a  profit 
of  37i%.     What  was  the  cost  of  the  land  ? 

32.  A  merchant  gained  in  one  year  $3650  on   goods 
sold  at  a  profit  of  20%.     What  was  the  cost  of  the  goods  ? 

33.  A  man  bought  a  farm  for  $97.75  per  acre,  which 
price  was  15%  more  than  was  previously  paid  for  it.    What 
was  the  previous  price  per  acre  ? 

SOLUTION.      The  purchase  price  =  115%  of  previous  price. 
.-.  115  %  of  previous  price  =  $  97.75. 
1  %  of  previous  price  =  $  .85. 
.-.  The  previous  price  =  $  85.00. 

34.  A  stationer  lost  22%  by  selling  paper  at  $2.925  per 
ream.     What  did  it  cost  him  ? 

SOLUTION.         The  selling  price  =  78  %  of  the  cost. 
.-.  78  %  of  the  cost  =  $  2.925. 
1  %  of  the  cost'=  $  .0375. 
.-.  The  cost  =  $3.75. 

.    35.    A  farmer  sold  a  cow  for  $44.50  and  gained  25%. 
What  was  the  cost  of  the  cow  ? 


250  PERCENTAGE. 

36.  A  city  lot  was  sold  for  $  1260,  which  was  an  advance 
of  12%  on  its  cost.     What  did  it  cost  ? 

37.  A  horse  and  a  wagon  were  sold  for  $  235,  at  a  gain 
of  10£%.    What  did  they  cost  ? 

38.  A  merchant  sold  a  bill  of  goods  for  $  136.44,  thereby 
gaining  20%.     What  did  the  goods  cost  him  ? 

39.  A  carriage  was  sold  for  $  361,  which  was  a  loss  of 
5%.     What  was  the  cost  ? 

40.  A  dealer  sold  some  cattle  for  $  980.28,  thereby  losing 
'  10  % .     What  did  they  cost  him  ? 

41.  At  a  forced  sale  a  house  was  sold  for  $4527,  which 
was  a  loss  of  33^%.     What  was  its  cost  ? 

42.  A  stock  of  goods  was  sold  for  $4575,  which  was  a 
loss  of  3£%.     What  did  the  goods  cost  ? 

43.  A  bookseller  bought  books  at  12 \°/0  discount  from 
the  retail  price,  which  was  $  2  per  volume,  and  sold  them 
at  the  retail  price.     What  was  his  gain  per  cent  ? 

44.  A  farmer  bought  80  acres  of  land  at  $  50  per  acre, 
and  spent  $  1800  for  improvements.     How  must  he  sell  it 
per  acre  so  as  to -gain  15%  ? 

45.  A  merchant  marked  cloth  at  25%  advance  on  the 
cost.     The  goods  being  damaged,  he  was  obliged  to  take  off 
20%  of  the  marked  price,  selling  it  at  $1  per  yard.     What 
was  the  cost  ? 

46.  Mr.  H  sold  two  houses  for  $3600  each.     On  one  he 
gained  25%,  and  on  the  other  he  lost  25%.     How  much 
was  gained  or  lost  by  the  transaction  ? 

47.  What  per  cent  is  gained  in  buying  coal  by  the  long 
ton,  at  $  4.50  a  ton,  and  selling  it  by  the  short  ton,  at  the 
same  price  ? 


COMMISSION.  251 

• 

COMMISSION. 

293.  1.  An  agent  sells  $1000  worth  of  goods.   How  much 
will  he  receive  for  his  services  if  he  gets  2%  of  the  sales  ? 

2.  If  an  agent  purchases  $  3000  worth  of  silks,  how  much 
ivill  he  receive  for  his  services  if  he  gets  3%  of  the  cost  of 
the  goods  ? 

3.  A  planter  paid  his  agent  5%  of  the  sum  received  for 
his  cotton.     If  the  cotton  was  sold  for  $  5000,  how  much 
did  the  agent  receive  for  his  services,  or  how  much  was  his 
commission  ? 

4.  At  2%  commission,  what  will  a  man  receive  for  selling 
property  to  the  value  of  $  800  ?     How  much  will  be  left 
after  paying  the  commission,  or  how  much  will  be  the  net 
proceeds  ? 

5.  If  2%  commission  is  paid  for  buying  goods,  what  is 
the  cost  of  every  dollar's  worth  of  goods  bought  ?     Since 
every  dollar's  worth  of  goods  bought  costs  the  purchaser 
$1.02,  how  many  dollars'  worth  can  be  bought  for  $  102  ? 
For  $  204  ? 

6.  How  many  dollars'  worth  of  goods  can  be  bought  for 
$618,  after  making  allowance  for  the  agent's  commission 
of  3%  ? 

294.  A  person  who   buys  or   sells  goods,  or  transacts 
business  for  another,  is  called  a  Commission  Merchant,  or 
Agent,  or  Broker. 

295.  The  compensation  allowed  a  commission  merchant 
is  called  his  Commission  or  Brokerage. 

296.  The  merchandise  sent  to  a  commission  merchant  to 
be  sold  is  called  a  Consignment. 

297.  The  person  who  sends  the  merchandise  is  called  the 
Consignor. 


252  PERCENTAGE. 

298.  The  person  to  whom  the  merchandise  is  sent  is 
called  the  Consignee. 

299.  The  sum  left  after  the  commission  and  expenses 
have  been  paid  is  called  the  Net  Proceeds. 

300.  PRINCIPLE.  —  The  commission  is  reckoned  at  a  certain 
rate  per  cent  upon  the  value  of  the  sales  and  purchases. 

WRITTEN    EXERCISES. 

301.  1.    What  will  be  an  agent's  commission  for  selling 
$  696  worth  of  goods  at  2f  %  ? 

SOLUTION.  —  2£%  or  .02f  of  $696  =  $19.14,  the  commission. 

2.  If  I  send  my  agent  $4590  to  invest  in  goods,  after 
deducting  his  commission  of  2%,  how  much  will  he  invest 
in  goods  for  me  ? 

SOLUTION.  —  Since  the  agent  receives  a  commission  of  2  %  for  his 
services,  it  requires  $1.02  to  purchase  $  1  worth  of  goods.  Therefore, 
he  can  purchase  as  many  dollars'  worth  of  goods  for  me  as  $1.02  is 
contained  times  in  $4590,  or  $4500  worth. 

3.  An  agent  sold  goods  to  the  amount  of  $1260.     What 
was  his  commission  at  3J%  ? 

4.  What  is  the  commission  at  2^%  for  selling  680  bu.  of 
wheat  at  $1.10  per  bushel  ? 

5.  An  agent  collects  $3450.     How  much  is  remitted  to 
the  employer  after  deducting  5%  commission. 

6.  A  real  estate  agent  was  paid  $  375  for  collecting  rents. 
How  much  did  he  collect,  his  commission  being  5%  ? 

7.  If  $7415  which  I  send  my  agent  includes  what  he  is 
to  invest  in  wheat  for  me  and  his  commission  of  2%  on  the 
purchase,  how  many  bushels  will  I  get,  when  wheat  is  worth 
$  .85  per  bu.  ? 

8.  My  agent  has  bought  585  bbl.  of  flour  at  $4.50  per 
barrel.     His  commission  is  2|-%.     How  much  money  must 
I  remit  to  pay  the  cost  of  the  flour  and  the  commission  ? 


COMMERCIAL  DISCOUNT.  253 

9.  A  commission  merchant  sold  goods  to  the  amount 
of  $4800,  charging  3|%  commission.  After  paying  $25 
charges,  he  invested  the  balance  in  raw  material,  and 
charged  2^%  for  the  investment.  How  much  was  invested? 

10.  A  speculator  sent  $  14,616  to  his  agent  in  Chicago, 
which  he  directed  him  to  invest  in  wheat.    After  deducting 
l-J-%  commission,  how  many  bushels  of  wheat  did  he  buy 
at  $.90 'a  bushel? 

11.  A  lawyer  collected  80%  of  a  debt  of  $2360,  and 
charged  5%  commission  on  the  sum  collected.     How  much 
did  the  creditor  receive  ? 

12.  A  merchant  sent  his  agent  in  New  Orleans  $  3536.25 
to  be  expended  in  cotton  after  deducting  his  commission  of 
2J%.     How  much  was  invested  in  cotton  ? 

13.  A  commission  merchant  having  sold  a  consignment 
of  cotton  for  $  2560,  retained  $100  to  pay  freight  charges 
amounting  to  $  10.40  and  his  own  commission.     What  rate 
per  cent  commission  did  he  charge  ? 

COMMERCIAL  DISCOUNT. 

302.  A  deduction  from  the  price  or  value  of  anything  is 
called  a  Commercial  Discount. 

303.  Manufacturers   and  wholesale   dealers   issue  price 
lists,  from  which  prices  various  discounts  are  allowed. 

Sometimes  several  discounts  are  allowed  the  purchaser.  In  such 
cases,  the  first  discount  is  to  be  deducted,  then  the  second  is  to  be 
computed  upon  the  remainder  and  deducted,  and  so  on  for  each  suc- 
cessive discount. 

Thus,  when  the  discounts  allowed  are  50  %,  10%,  and  5  %,  the  50  %  is 
first  deducted ;  from  the  remainder  10  %  is  deducted ;  and  from  that 
remainder  5%  is  deducted. 

304.  The  amount  of  a  bill  less  the  discounts  is  called 
the  Net  Amount. 


254  PERCENTAGE. 

WRITTEN    EXERCISES. 
305.   Find  the  net  amounts  of  the  following  bills : 

1.  $450,  discounts,  20%  and  10%. 

2.  $760,  discounts,  20%  and  15%. 

3.  $840,  discounts,  25%  and    5%. 

4.  $976,   discounts,  35%  and  10%. 

5.  $8.75,  discounts,  10%  and  10%. 

6.  $6.80,  discounts,  40%  and    5%. 

7.  $350,   discounts,  30%  and  15%. 

8.  $375.20,  discounts,  10%,  10%,  and  10%. 

9.  $280.50,  discounts,  15%,  10%,  and  5%. 

10.  Find  the  net  amount  of  a  bill  of  $520,  the  discounts 
being  15%  and  10%. 

11.  What  is  the  difference  on  a  bill  of  $320,  between  a 
direct  discount  of  35%  and  successive  discounts  of  20% 
and  15%? 

12.  What  is  the  net  amount  of  a  bill  of  $60.80,  the  dis- 
counts being  30%  and  10%  ? 

13.  What  is  the  net  amount  of  a  bill  of  goods,  the  list 
price  of  which  is  $345,  trade  discount  8%,  and  5%  off  for 
cash? 

14.  A  bill  of  goods  at  list  prices  amounted  to  $420.65. 
The  discounts  were  25  %  and  10  % .   What  was  due  on  the  bill? 

15.  On  a  bill  of  goods  amounting  to  $230,  what  is  the 
difference  between  a  discount  of  45%  and  successive  dis- 
counts of  25%,  15%,  and  5%  ? 

16.  What  is  the  net  amount  of  a  bill  of  $  450,  discounts 
being  30%,  10%,  and  5%  ? 

17.  What  is  the  net  amount  of  a  bill  of  $360,  discounts 
being  12|%  and  8%  ?     Find  a  single  discount  equivalent 
to  these  two  successive  discounts. 


TAXES.  255 

TAXES. 

306.  1.    If  a  man  pays  annually  for  public  purposes  1% 
of  the  value  of  his  property,  estimated  at  $  10,000,  what  is 
the  amount  of  his  tax  f 

2.  If  I  am  taxed  1^%  on  land,  houses,  etc.,  or  real  estate 
estimated  at  $  8000,  what  is  the  amount  of  my  tax  ? 

3.  If  a  man  is  taxed  1|%  on  his  money,  mortgages,  cat- 
tle, or  personal  property ,  estimated  at  $  5000,  what  is  the 
amount  of  his  tax  ? 

307.  Fixed  property,  as  land,  etc.,  is  called  Real  Estate. 

308.  Movable  property,  as  money,  mortgages,  cattle,  lum- 
ber, etc.,  is  called  Personal  Property. 

309.  A  sum  of  money  assessed  upon  the  persons,  prop- 
erty, or  business  of  individuals  is  called  a  Tax. 

1.  A  tax  upon  property  is  reckoned  at  a  certain  rate  per  cent  upon 
the  estimated  or  assessed  value  of  the  property. 

2.  A  tax  upon  the  person  is  a  fixed  sum  assessed  upon  each  person. 
It  is  called  a  Poll  or  Capitation  Tax. 

Non-resident  tax-payers  are  not  subject  to  a  poll  tax. 

310.  The  officers  appointed  to  estimate  -the  taxable  value 
of  property  are  called  Assessors.     They  receive  a  salary. 

The  officer  appointed  to  collect  the  taxes  is  called  a  Col- 
lector. He  receives  either  a  salary  or  a  per  cent  of  the  tax 
collected. 

311.  A  list  of  the  names  of  the  taxable  inhabitants,  with 
the  assessed  valuation  of  each  person's  property  and  the 
amount  of  his  tax,  is  termed  an  Assessment  Roll. 

312.  Before  taxes  are  assessed,  a  complete  inventory  of 
all  the  taxable  property  must  be  made. 

If  the  assessment  includes  a  poll  tax,  a  complete  list  of 
all  the  taxable  polls  must  also  be  made  out 


256 


PERCENTAGE. 


WRITTEN    EXERCISES. 

313.  1 .   A  village  must  raise  $  8795  on  property  assessed 
at  $989,387,  and  there  are  670  persons  subject  to  a  poll  tax 
of  $  1  each.     A's  property  is  assessed  at  $  10,000,  and  he  is 
a  resident  of  the  village.     What  will  be  his  tax  ? 

SOLUTION. 

$8795  —  $  670  =  $  8125,  amount  to  be  levied  upon  property. 
$8125  -T-  $989,387  =  .00821,  or  8T^  mills  on  $1. 
$10,000  x  .00821  =  $82.10,  A's  property  tax. 
$   1.00,  A' spoil  tax. 

$83.10,  A's  entire  tax. 

314.  To  facilitate  the  computation  of  taxes,  assessors 
usually  prepare  a  table  like  the  following.     The  rate  at 
which  this  table  is  computed  is  .00821. 


PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

$1 

$.00821 

$10 

$.0821 

$100 

$  .821 

$1000 

$  8.21 

2 

.01642 

20 

.1642 

200 

1.642 

2000 

16.42 

3 

.02463 

30 

.2463 

300 

2.463 

3000 

24.63 

4 

.03284 

40 

.3284 

400 

3.284 

4000 

32.84 

5 

.04105 

50 

.4105 

500 

4.105 

5000 

41.05 

6 

.04926 

60 

.4926 

600 

4;  926 

6000 

49.26 

7 

.05747 

70 

.5747 

700 

5.747 

7000 

57.47 

8 

.06568 

80 

.6568 

800 

6.568 

8000 

65.68 

9 

.07389 

90 

.7389 

900 

7.389 

9000 

73.89 

2.   Find  B's  tax  upon  property  assessed  at  $8550. 
•   SOLUTION. 


Tax,  by  table  on  $8000  =  $65.68 
"    -"      »      »       500=      4.105 
«      "      "      "          50  =        .4105 


u     u  $8550  =  $70. 1955,  B's  property  tax. 


DUTIES  OR  CUSTOMS.  257 

3.  Find  C's  tax  upon  property  assessed  at  $  5780. 

4.  What  is  D's  tax,  whose  property  is  assessed  at  $12,650, 
and  who  pays  for  2  polls  ? 

5.  What  is  E's  tax,  whose  property  is  assessed  at  $6759, 
and  who  pays  for  3  polls  ? 

6.  What  is  the  tax  on  property  valued  at  $  13^417.40  ? 

7.  In  a  town  containing  390  polls,  assessed  at  $1  each, 
the  assessment  roll  shows  the  valuation  of  the  property  to 
be  $987,680.     The  amount  of  tax  to  be  raised  is  $5822.24 
What  is  the  rate  of  taxation  ? 

8.  At  the  rate  of  $  9.50  on  $1000,  find  the  tax  paid  by 
a  man  who  pays  a  poll  tax  of  $  1.25,  and  whose  property  is 
valued  at  $  19,430. 

9.  The  taxable  property  in  a  certain  town  is  valued  at 
$  1,360,000,  and  a  tax  of  $  8840  is  voted  for  school  purposes. 
What  is  the  rate  of  taxation  ? 

10.  In  the  same  town  A's  property  is  assessed  at  $  3150, 
B's  at  $  4200,  and  C's  at  $  5595.     How  much  tax  is  each 
required  to  pay  ? 

11.  If  the  rate  of  taxation  is  7-^- mills  on  a  dollar,  and  the 
tax  on  a  farm  is  $48.15,  what  is  its  assessed  value  ? 

DUTIES  OR  CUSTOMS. 

315.  Taxes   levied  by  the   government  upon  goods  im- 
ported from  other  countries  are  termed  Duties. 

316.  When  the  duty  is  a  certain  per  cent  of  the  cost  of 
the  goods  it  is  called  an  Ad  Valorem  Duty. 

317.  When  the  duty  is  a  fixed  tax  upon  an  article  with- 
out regard  to  ite  value,  it  is  called  a  Specific  Duty. 

318.  An  inventory  or  list  of  the  goods,  and  the  prices  at 
which  they  were  purchased,  is  termed  an  Invoice  or  Manifest. 

STAND.    AR. —  17 


258  PERCENTAGE. 

319.  Before  computing  the  duties  on  certain  classes  of 
merchandise,  allowances  are  made  for  tare,  or  the  weight  of 
the  box,  bag,  etc.,  for  leakage,  breakage,  etc. 

WRITTEN    EXERCISES. 

320.  1.   What  is  the  duty  on  420  yards  of  broadclot^ 
invoiced  at  $  1.75  per  yard,  at  25%  ad  valorem  ? 

2.  What  is  the  duty,  at  2^  per  pound,  on  2800  Ib.  of 
rice,  allowing  5%  for  tare  ? 

3.  A  merchant   imported   goods  invoiced  at  £530  5s. 
What  was  the  duty  at  45%  ad  valorem  ? 

4.  What  was  the  cost  per  dozen  of  6  gross  of  penknives 
invoiced  at  $638.24  if  the  duty  was  40%  ad  valorem  ? 

5.  A  merchant   imported   1560   yards   of   Irish   linen, 
invoiced  at  38^  per  yard.     What  was  the  duty  at  35%  ad 
valorem  ? 

6.  What  is  the  duty,  at  25%  ad  valorem,  on  280  chests 
of  tea,  each  containing  60  pounds,  invoiced  at  45^  a  pound  ? 

7.  What  is  the  duty,  at  2%f  a  pound,  on  36  boxes  of 
raisins,  each  weighing  24  Ib.,  tare  5J  Ib.  a  box? 

8.  What  is  the  duty,  at  33^  %  ad  valorem,  on  45  tons  of 
steel,  of  2240  Ib.  each,  invoiced  at  5J^  per  pound  ? 

9.  What  is  the  duty,  at  25%  ad  valorem,  on  80  dozen 
watch  crystals,  invoiced  at  $  1.50  a  dozen,  an  allowance  of 
5%  being  made  for  breakage  ? 

10.  What  is  the  duty  on  18  pieces  of  Brussels  carpeting, 
of  60  yd.  each,  invoiced  at  45^  per  yd.,  the  specific  duty 
being  38^  per  yd.,  and  the  ad  valorem  duty  35%  ? 

11.  Find  the  duty  paid  on  the  following  importation: 
320  Ib.  of  knit-goods,  valued  at  $  1225,  at  35^  per  Ib.  spe- 
cific duty  and  40%  ad  valorem;   120  yd.  of  silk  invoiced 
at  $1.25  per  yard,  at  50%  ad  valorem;  and  500  yd.  of  lace, 
invoiced  at  87^  per  yard,  at  40%  ad  valorem. 


INSURANCE.  259 

INSURANCE. 

321.  1.   How  much,  will  it  cost  to  secure  myself  against 
loss  by  fire,  or  to  insure  my  property  for  $  6000,  if  an 
annual  sum  or  premium  of  1%  is  charged  by  those  who 
assume  the  risk  ? 

2.  What  will  be  the  cost  of  insuring  a  building  worth 
$  10,000,  at  |%,  for  £  of  its  value  ? 

3.  A  merchant  insured  a  stock  of  goods  worth  $12,000 
for  J  of  its  value,  at  1%.     What  was  his  annual  premium  ? 

4.  I  paid  an  annual  premium  of  $75  for  insuring  my 
property  at  f  %.     For  how  much  was  it  insured  ? 

322.  Indemnity  against  loss  or  damage  is  termed  Insur- 
ance.    Insurance  is  of  two  kinds,  Property  Insurance  and 
Personal  Insurance. 

323.  The  contract  between  the  insurance  company  and 
the  person  insured  is  called  the  Policy. 

324.  The  sum  paid  for  insurance  is  called  the  Premium. 

1.  A  company  in  which  the  person  insured  participates  in  the  profits 
and  shares  the  losses,  is  called  a  Mutual  Insurance  Company. 

2 .  Many  mutual  companies  charge  fixed  rates  of  premium,  and  return 
to  each  policy  holder  annually  his  share  of  the  surplus. 

3.  Another  kind  of  mutual  company  assesses  upon  each  person 
insured  his  share  of  the  loss,  whenever  a  loss  occurs.     Such  com- 
panies are  sometimes  called  Assessment  Companies. 

4.  A  company,  in  which  the  capital  to  meet  the  losses  is  contributed 
by  stockholders  who  alone  share  in  the  profits  and  losses  of  the  busi- 
ness, is  called  a  Stock  Company. 

PROPERTY  INSURANCE. 

325.  Property  Insurance  includes  indemnity  against  loss 
or  damage  by  fire,  or  Fire  Insurance;  against  loss  or  damage 
by  casualties  at  sea,  or  Marine  Insurance;  against  loss  or 
damage  to  cattle,  horses,  etc.,  or  Live  Stock  Insurance,  etc. 


260  PERCENTAGE. 

WRITTEN   EXERCISES. 

326.  1.   How  much  is  the  annual  premium  on  a  policy 
of  insurance  on  a  factory  for  $  8500,  if  the  rate  is  2f  %  ? 

2.  A  house  was  insured  for  $3600,  at  l-J-%.    What  was 
the  premium  ? 

3.  A  factory  worth  $45,000  is  insured  for  •§-  of  its  value, 
at  If  %.     How  much  is  the  premium  ? 

4.  A  man  insured  a  row  of  7  houses  at  $  5800  each,  pay- 
ing an  annual  rate  of  If  %.     How  much  does  it  cost  him  ? 

5.  If  a  premium  of  $75  is  paid   for   an   insurance  of 
$  6400  on  a  house,  what  is  the  rate  of  insurance  ? 

6.  A  cargo  worth  $9670  was  insured  for  -^  of  its  valne, 
at  3^%.     In  case  of  shipwreck,  what  would  be  the  actual 
loss  to  the  owner  ? 

7.  What  will  it  cost  to  insure  a  building  worth  $  9840 
for  |  of  its  value,  at  -J%  ? 

8.  A  dealer  paid  $375  for  the  insurance  of  a  cargo  of 
grain,  at  1J%.     What  was  the  amount  of  insurance  ? 

9.  A  merchant  sent  his  agent  in  St.  Paul  $3493.50  to 
invest  in  flour  at  $  4.25  per  barrel  after  deducting  a  commis- 
sion of  2f  %.     The  flour  was  insured  at  \\%,  and  $268.25 
was  paid  for  transportation.     If  the  flour  was  then  sold  at 
a  gain  of  10%  on  the  whole  cost,  what  was  the  selling  price 
per  barrel  ? 

PERSONAL  INSURANCE. 

327.  Indemnity  against  loss  of  life,  or  Life  Insurance; 
against  loss  occasioned  by  accidents,  or  Accident  Insurance; 
against  loss  occasioned  by  sickness,  or  Health  Insurance,  are 
varieties  of  Personal  Insurance. 

Of  these  the  most  important  kind  is  Life  Insurance. 


INSURANCE.  261 

328.  The  policies  issued  by  life  insurance  companies  are 
of  various  kinds,  the  chief  of  which  are  the  Life  Policy  and 
the  Endowment  Policy. 

1.  A  policy  which  secures  the  payment  of  a  sum  of  money  at  the 
death  of  the  person  insured  is  called  a  Life  Policy. 

2.  A  policy  which  secures  the  payment  of  a  sum  of  money  at  a 
specified  time  or  at  death,  if  it  occurs  before  the  specified  time,  is 
termed  an  Endowment  Policy. 


WRITTEN    EXERCISES. 

329.   1.  How  much  will  be  the  annual  premium  on  a  life 
insurance  policy  for  $4000,  at  $25.70  per  $1000? 

2.  What  is  the  annual  premium  on  a  life  policy  of  $  6500, 
at  $  29.50  per  $  1000  ? 

3.  A  man  paid  a  Mutual  Insurance  Company  for  30  years 
an  annual  premium  on  a  life  policy  for  $  3000,  of  $  26.30 
per  $1000.     Of  this  premium  15%  was  returned  as  divi- 
dends.    How  much  did  he  pay  in  all  ? 

4.  My  life  is  insured  for  $  9000,  at  an  annual  cost  of 
$315.     What  is  the  annual  premium  per  $  1000  ? 

5.  If  a  person  who  is  insured  for  $6000,  at  an  annual 
premium  of  $31.40  per  $1000,  dies  after  12  payments, 
how  much  more  will  his  heirs  get  than  has  been  paid  in 
premiums  ? 

6.  A  man  insured  his  life  for  $8000,  paying  $26.30  per 
$  1000.     If  he  should  live  20  years  after  he  was  insured, 
what  would  be  the  amount  of  the  premiums  paid  ? 

7.  A  man  at  the  age  of  35  secured  a  policy  upon  his  life 
for  $  5000,  paying  the  first  year  $  172.50,  which  included  $  1 
for  examination.    What  was  the  premium  paid  upon  $  1000  ? 


INTEREST. 


330.  1.   When  a  sum  equal  to  6%  of  the  money  loaned 
is  paid  for  the  use  of  it  for  1  yr.,  how  much  must  be  paid 
for  the  use  of  $  100  for  1  yr.  ?    For  2  yr.  ?     For  3  yr.  ? 

2.  When  the  sum  paid  for  the  use  of  money  is  10% 
annually,  what  must  be  paid  for  the  use  of  $  240  for  1  yr.  ? 
For  6  mo.  ?     For  3  mo.  ?     For  1  mo.  ?     For  £  mo.  ?     For 
10  da.,  or  -j-  of  a  month  ?     For  20  da.  ? 

3.  If  $500  is  loaned  for  2  yr.,  at  6%  per  year,  what 
will  be  the  amount  due  at  the  end  of  that  time  ? 

331.  The  sum.  paid  for  the  use  of  money  is  called  Interest. 

332.  The  sum  for  the  use  of  which  interest  is  paid  is 
termed  the  Principal. 

333.  The  sum  of  the  principal  and  interest  is  called  the 
Amount. 

334.  In  computing  interest  it  is  usual  to  regard  a  year  as 
12  months,  and  a  month  as  30  days. 

335    To  compute  interest. 

1.  What  is  the  interest  of  $100  for  1  yr.  at  5%  ?     For 

2  yr.  ?     For  3  yr.  ?    For  4  yr.  ?     For  10  yr.  ? 

2.  What  is  the  interest  of  $200  for  2  yr.  at  6%  ?     For 

3  yr.  ?     For  1£  yr.  ?     For  2£  yr.  ?     For  1  yr.  6  mo.  ? 

3.  What  is  the  interest  of  $400  for  1  yr.  at  4%  ?     For 
1£  yr.  ?     For  1£  yr.  ?     For  1  yr.  3  mo.  ?     For  1  yr.  6  mo.  ? 
For  1  yr.  8  mo.  ? 

262 


WRITTEN   EXERCISES.  263 

WRITTEN   EXERCISES. 

336,     1.   What  is  the  interest  of  $375.15  for  4  yr.  at 

5%  ? 

SOLUTION. 

$375.15,  Principal. 
.05,  Rate. 

$18.7575,  Interest  for  1  yr. 
4 

$75.0300,  Interest  for  4  yr. 
Find  the  interest  of  : 

2.  $ 496.84  for  5  yr.  at  8%.     6.    $  250.50  for    4  yr.  at  6%. 

3.  $389.50  for  4  yr.  at  7%.     7.    $375.65  for    7  yr.  at  &%. 

4.  $  541.76  for  6  yr.  at  5%.     8.   $  460.70  for    5  yr.  at  7%. 

5.  $  756.38  for  7  yr.  at  8%.      9.    $ 695.49  for  10  yr.  at  6%. 

10.   Find  the  interest  of  $453.20  for  2  yr.  8  mo.  at  5%. 
SOLUTION. 

$453.20 
.05 

$22.6600,  Int.  for  1  yr. 
2 


$45.3200,  Int.  for  2  yr. 

%  of  the  int.  for  1  yr.  =    11.3300,  Int.  for  6  mo. 

|  of  the  int.  for  6  mo.  =      3.7766,  Int.  for  2  mo. 

$60.4266,  Int.  for  2  yr.  8  mo. 
Find  the  interest  of  : 

11.  $687.35  for  3  yr.  6  mo.  at  6%. 

12.  $476.38  for  4  yr.  8  mo.  at  6%. 

13.  $380.40  for  4  yr.  5  mo.  at  1%. 

14.  $425.60  for  5  yr.  7  mo.  at  5%. 

15.  $368.52  for  6  yr.  4  mo.  at  9%. 

16.  $410.30  for  7  yr.  3  mo.  at  1%. 


264  INTEREST. 

17.  $564.80  for  Syr.  5  mo.  at  8%. 

18.  $  672.50  for  5  yr.  7  mo.  at  4%.. 

19.  $  150.18  for  2  yr.  6  mo.  at  6%. 

20.  $  175.40  for  3  yr.  7  mo.  at  5%. 

21.  $233.50  for  2  yr.  5  mo.  at  8%. 

22.  $  317.42  for  4  yr.  3  mo.  at  7%. 

23.  $  510.12  for  4  yr.  8  mo.  at  5%. 

24.  $  468.72  for  5  yr.  6  mo.  at  6%. 

25.  $496.88  for  6  yr.  9  mo.  at  8%. 

26.  $  784.75  for  7  yr.  8  mo.  at  9%. 

27.   Find  the  amount  of  $240.15  for  2  yr.  5  mo.  13  da.  at 

6%. 

SOLUTION. 

$240.15 
.06 


$  14.4090,  Int.  for  1  yr. 
2 

$28.8180,  Int.  for  2  yr. 

|  of  the  int.  for  1  yr.  =  4.8030,  Int.  for  4  mo. 

£  of  the  int.  for  4  mo.=   1.2007,  Int.  for  1  mo. 

$  of  the  int.  for  1  mo.=     .4002,  Int.  for  10  da. 

-J-  of  the  int.  for  10  da.=     .0800,  Int.  for  2  da. 

\  of  the  int.  for  2  da.  =     .0400,  Int.  for  1  da. 

$   35.3419,  Int.  for  2  yr.  5  mo.  13  da. 
$240.15     ,  Principal. 

$275.49    ,  Amount. 

It  is  customary  to  disregard  the  mills  if  they  are  less  than  5,  and 
to  call  them  1  cent  if  they  are  more  than  5. 

The  method  of  computing  interest  illustrated  above  is  called  the 
method  by  Aliquot  Parts  or  the  Business  Method. 

Find  the  interest  and  amount  of : 

28.  $313.50  for  2  yr.  3  mo.  at  6%. 

29.  $935.75  for  3  yr.  5  mo.  at  7%. 


WRITTEN  EXERCISES.  266 

30.  $269.50  for  2  yr.  7  mo.  10  da.  at  5%. 

31.  $468.75  for  1  yr.  5  mo.  15  da.  at  1%. 

32.  $274.08  for  2  yr.  7  mo.    5  da.  at  6%. 

33.  $364.50  for  2  yr.  8  mo.  20  da.  at  5%. 

34.  $286.09  for  3  yr.  5  mo.  10  da.  at  8%.. 

35.  $368.75  for  Syr.  8  mo.  15  d'a.  at  7%. 

36.  $368.18  for  4  yr.  6  mo.  12  da.  at    6%. 

37.  $580.90  for  5  yr.  8  mo.  18  da.  at  8%. 

38.  $275.60  for  1  yr.  9  mo.  15  da.  at  5%.. 

39.  $468.25  for  2  yr.  7  mo.  11  da.  at  6%. 

40.  $  815.27  for  3  yr.  8  mo.  21  da.  at  10%.. 

41.  $  125.80  for  2  yr.  4  mo.  10  da.  at  5%. 

42.  $  184.50  for  1  yr.  2  mo.  18  da.  at  8%. 

43.  $  560.25  for  3  yr.  5  mo.  10  da.  at  7%. 

44.  $376.47  for  2  yr.  9  mo.  13  da.  at  6%. 

45.  $1000      for3yr.  7  mo.  21  da.  at  7%. 

46.  $4120      for  5  yr.  3  mo.  18  da.  at  5%. 

47.  $3180     for  2  yr.  10  mo.  16  da.  at  4%. 

48.  $2875      for  4  yr.  11  mo.  17  da.  at  6%. 

Find  the  amount  of : 

49.  $  685.20  from  June  12,  1892,  to  Aug.  10, 1893,  at  6%. 

yr.        mo.     da. 

1893     8     10  SOLUTION. — The  time  is  1  yr.  1  mo.  28  da.,  and 

1892     6     12      the  amount'  °f  $685.20  for  the  time  and  rate  is 

1  1    28     *732'94' 
Find  the  amount  of : 

50.  $  423.36  from  June  7, 1890,  to  Dec.  15,  1891,  at  6%. 

51.  $346.85  from  Sept.  10,  1891,  to  Apr.  8,  1892,  at  5%. 

52.  $  427.93  from  Apr.  23,  1892,  to  Nov.  13, 1892,  at  6%. 

53.  $  684.14  from  Oct.  15,  1891,  to  Mar.  8,  1892,  at  7%. 

54.  $  713.62  from  Nov.  18, 1892,  to  Apr.  13, 1893,  at  8%. 


266  INTEREST. 

337.  The  six  per  cent  method. 

This  method  is  very  convenient  because  of  the  ease  with 
which  the  interest  of  $  1  can  be  computed.     Thus, 

The  interest  of  $  1  for  1  yr.    =  $  .06. 
The  interest  of  $  1  for  1  mo.  =  $  .005. 
The  interest  of  $1  for  6  da.  =  $.001. 
The  interest  of  $  1  for  1  da.   =  $  .000£. 

1.    What  is  the  interest  of  $  480.60  for  3  yr.  4  mo.  12  da. 

at  6%  ? 

SOLUTION. 

The  interest  of  $  1  for    3  yr.  =  $  .18 

The  interest  of  $  1  for    4  mo.  =    .02 

The  interest  of  $  1  for  12  da.  =     .002 

The  interest  of  $  1  for    3  yr.   4  mo.  12  da.  =  $  .202 
The  interest  of  $480.60  =  $.202  x  480.60,  or  $97.08. 

What  is  the  interest  of : 

2.  $  575.40  for  2  yr.  2  mo.    6  da.  at  6%  ? 

3.  $434.70  for  3  yr.  4  mo.  18  da.  at  6%  ? 

4.  $387.62  for  2  yr.  6  mo.  12  da.  at  6%  ? 

5.  $292.47  for  3  yr.  8  mo.  24  da,  at  6%  ? 

6.  $436.45  for  4  yr.  7  mo.  15  da.  at  6%  ? 

7.  $  672.36  for  1  yr.  9  mo.  21  da.  at  6%  ? 

8.  $  945.50  for  3  yr.  4  mo.  18  da.  at  6%  ? 

9.  $  392.00  for  5  yr.  7  mo.  24  da.  at  6%  ? 

338.  When  the  interest  of  any  sum  of  money  at  6%  has 
been  found,  the  interest  of  the  same  sum  at  7%  may  be 
found  by  adding  1  of  the  interest  to  the  result;  at  8%  by 
adding  -J  of  the  interest,  etc.     Consequently,  the  6%  method 
may  be  used  to  compute  interest  at  any  rate. 


WRITTEN   EXERCISES.  267 

What  is  the  interest  of : 

10.  $280.75  for  3  yr.  2  mo.  12  da.  at  7%  ? 

11.  $  315.40  for  5  yr.  7  mo.  18  da.  at  8%  ? 

12.  $416.26  for  8  yr.  9  mo.  15  da.  at  5%  ? 

13.  $620.35  for  7  yr.  5  mo.  19  da.  at  9%  ? 

14.  $  575.38  for  9  yr.  7  mo.  13  da.  at  4%  ? 

339.   Method  by  days. 

When  the  time  is  short,  interest  is  computed  for  the 
actual  number  of  days,  considering  a  year  as  360  days. 

1.  What  is  the  interest  of  $108.12  from  March  15  to 
June  10  at  6%  ? 

SOLUTION.  — From  March  15  to  June  10  is  87  days.  The  interest 
of  $  1  for  87  da.  at  6  %  is  $  .014£.  Therefore  the  interest  of  $  108.12  is 
$.014^x108.22,  or  $1.57;  or, 

$108.12  ='the  principal. 

1  %  of  principal  =  $  1 . 0812 ,  Int.  for  60  da. 
$  of  int.  for  60  da.  =  .3604,  Int.  for  20  da. 
|  of  int.  for  20  da.  =  .0901,  Int.  for  5  da. 
Jt  of  int.  for  20  da.  =  .03604,  Int.  for  2  da. 

$1.56774,  Int.  for  87  da. 

In  computing  the  number  of  days,  reckon  from  the  day  upon  which 
the  sum  was  loaned,  and  include  the  day  upon  which  it  was  paid. 

Find  the  interest  of : 

2.  $840  from  Mar.  1,  1891,  to  May  5,  1891,  at  6%. 

3.  $950  from  Feb.  3,  1891,  to  Apr.  15,  1891,  at  6%. 

4.  $879  from  Jan.  18,  1891,  to  June  1,  1891,  at  6%. 

5.  $895  from  Mar.  20,  1891,  to  Aug.  18,  1891,  at  6%. 

6.  $965  from  May  25,  1891,  to  Oct.  10,  1891,  at  6%. 

7.  $  1050  from  Apr.  1,  1891,  to  July  22,  1891,  at  6%. 

8.  $1120  from  Jan.  1,  1892,  to  Mar.  8,  1892,  at  6%. 

9.  $3000  from  Feb.  2,  1892,  to  Sept.  8,  1892,  at  6%. 


268  INTEREST. 

340.  The  method  by  months. 

1.    What  is  the  interest  of  $  290.75  for  2  yr.  3  mo.  21  da. 

at7%? 

SOLUTION. 

21  da.=f£  or  ^  of  a  month.    Therefore  2  yr.  3  mo.  21  da.=27.7  mo. 

$290.75 

.07 

12)$20.3525,  Int.  for  1  yr. 
$1.6960,  Int.  for  1  mo. 

27.7 
$46.9792,  Int.  for  27.7  mo.,  or  2  yr.  3  mo.  21  da. 

What  is  the  interest  of: 

2.  $  270.60  for  1  yr.  2  mo.  15  da.  at  7%  ? 

3.  $285.45  for  3  yr.  4  mo.  20  da.  at  5%  ? 

4.  $315.65  for  2  yr.  3  mo.  10  da.  at  6%  ? 

5.  $  573.95  for  6  yr.  5  mo.  24  da.  at  8%  ? 

6.  $397.85  for  2  yr.  3  mo.    5  da.  at  6%  ? 

7.  $463.28  for  1  yr.  2  mo.    7  da.  at  7%  ? 

8.  $395.18  for  3  yr.  4  mo.    8  da.  at  5%  ? 

9.  $  793.64  for  5  yr.  6  mo.    9  da.  at  4%  ? 

341.  To  compute  accurate  interest. 

As  has  been  said,  a  year  is  usually  considered  as  12 
months  of  30  days  each,  or  360  days,  when  the  time  is  less 
than  a  year  and  expressed  in  months  and  days.  Some- 
times, however,  the  interest  is  computed  for  the  exact 
number  of  days,  365  days  constituting  a  year. 

Accurate  interest  for  a  given  number  of  days  will,  there- 
fore, be  that  number  of  365ths  of  the  interest  for  1  yr.,  or 
the  ordinary  interest  diminished  by  -g^-,  or  ^  of  itself. 

Find  the  accurate  interest  at  6  %  of : 

1.  $  650.25  for  61  da.  4.  $  920.40  for  82  da. 

2.  $  785.75  for  58  da.  5.  $ 3296.80  for  110  da. 

3.  $  872.90  for  73  da.  6.  $  5375.80  for  295  da, 


ANNUAL  INTEREST.  269 

ANNUAL  INTEREST. 

342.  Simple  interest  upon  the  principal,  and  upon  any 
interest  overdue,  is  called  Annual  Interest. 

1.  The  contract  should  contain  the  words  "interest  payable  annu- 
ally" or  "annual  interest." 

2.  In  some  States  annual  interest  is  illegal. 

WRITTEN   EXERCISES. 

343.  1.   Find  the  amount  of  $  3500  for  4  yr.  6  mo.,  with 
interest  payable  annually  at  6%. 

SOLUTION. 

Int.  of  $  3500  for  4£  yr.  =  $  945.00 

The  annual  interest  is  $210. 

The  interest  for  the  first  yr.  remains  unpaid  for  3£  yr. ; 
the  interest  for  the  second  yr.  2£  yr. ,  etc.  There- 
fore the  unpaid  interest  drew  interest  for  3$,  2£, 
1£,  and  ^  yr.,  or  8  yr.,  and  the  interest  upon  $210 
for  that  time  is  100.80 

/.  The  entire  interest  due  is  $  1045.80 

$3500  +  $  1045.80  =  $4545.80,  Amt. 

Find  the  amount  of  the  following  with  annual  interest : 

2.  $  1200  for  3  yr.  4  mo.  at  6%. 

3.  $1420  for  4  yr.  6  mo.  at  6%. 

4.  $  1825  for  5  yr.  8  mo.  at  7%. 

5.  $  1976  for  3  yr.  6  mo.  12  da.  at  6%. 

6.  $2300  for  3  yr.  5  mo.  18  da.  at  8%. 

7.  $2760  for  5  yr.  3  mo.  6  da.  at  5%. 

8.  $3500  for  4  yr.  7  mo.  24  da.  at  9%. 

9.  $4100  for  3  yr.  5  mo.  15  da.  at  6%. 

10.  $  5450  for  4  yr.  8  mo.  24  da.  at  6%. 

11.  $  10,000  for  5  yr.  6  mo.  15  da.  at  5%. 

12.  $  7090  for  6  yr.  3  mo.  12  da.  at  6%. 


270  INTEREST. 


COMPOUND  INTEREST. 

344.  Interest  upon  the  principal  and  its  unpaid  interest 
combined  at  regular  intervals  is  Compound  Interest. 

1.  Interest  is  usually  compounded  annually,  semi-annually,  or  quar- 
terly, according  to  agreement. 

2.  Compound  interest  cannot  usually  be  enforced  by  law,  even  if  it 
is  specified  in  the  contract. 

3.  Most  savings  banks  allow  compound  interest  upon  balances 
remaining  on  deposit  for  a  full  interest  term. 

WRITTEN    EXERCISES. 

345.  1.  Find  the  compound  interest  of  $250  for  2  yr. 

3  mo.  at  6%. 

SOLUTION. 

$250         Principal. 

15         Interest  for  1st  yr.  at  6  %. 

Principal  for  2d  yr. 
Interest  for  2d  yr.  at  6  %. 

$280.90    Principal  for  3d  yr. 

4.21     Interest  for  3  mo.  at  6  %. 

$285.11     Amount  for  2  yr.  3  mo.  at  6  %. 
250         Original  principal. 

$35.11     Compound  interest  for  2  yr.  3  mo.  at  6  %. 

1.  Unless  it  is  specified  otherwise  in  the  agreement,  interest  is 
understood  to  be  compounded  annually.    ' 

2.  If  interest  is  compounded  semi-annually,  the  rate  must  be  con- 
sidered one  half  the  annual  rate,  if  quarterly,  one  fourth,  etc. 

3.  When  the  time  consists  of  years,  months,  and  days,  the  amount 
is  to  be  found  for  the  greatest  number  of  entire  periods,  as  years, 
half-years,  quarter-years,  etc.,  and  the  simple  interest  upon  this  for 
the  rest  of  the  time. 

Find  the  compound  interest  of  the  following  : 

2.  $ 275  for  2  yr.  2  mo.  at  6%. 

3.  $  310  for  3  yr.  6  mo.  at  7%. 

4.  $ 425  for  2  yr.  5  mo.  at  5%. 


COMPOUND  INTEREST.  271 

5.  $  650  for  2  yr.  9  mo.  at  6%. 

6.  $  535  for  3  yr.  5  mo.  at  7%. 

7.  $  580  for  3  yr.  8  mo.  20  da.  at  6%. 

8.  $260  for  2  yr.  6  mo.  at  6%,  payable  semi-annually. 

9.  $450  for  2  yr.  2  mo.  at  8%,  payable  quarterly. 

10.  What  is  the  compound  interest  of  $325.10  for  3  yr. 

2  mo.  at  6%  ? 

SOLUTION. 

By  referring  to  the  table  upon  the  next  page,  it  will  be  seen  that 
the  amount  of  $  1  for  3  yr.  at  6%  is  $  1. 191016.  Computing  the  interest 
upon  this  sum  for  2  mo.,  the  amount  of  $  1  for  3  yr.  2  mo.  is  $  1.202926. 

Therefore  the  amount  of  $325.10  is  325.10  times  that  sum. 

This  product  minus  the  principal  is  the  compound  interest. 

Find  the  compound  interest  of  the  following,  making  use 
of  the  table  : 

11.  $420.80  for  4  yr.  6  mo.  at  6%. 

12.  $430. 75  for  3  yr.  4  mo.  at  5%. 

13.  $  510.60  for  5  yr.  6  mo.  at  1%. 

14.  $  750.80  for  6  yr.  7  mo.  at  6%. 

15.  $  672.28  for  2  yr.  3  mo.  18  da.  at  6%. 
16     $856.57  for  4  yr.  8  mo.  10  da.  at  8%. 

17.  $889.37  for  6  yr.  9  mo.  21  da.  at  7%. 

18.  $  985.50  for  8  yr.  7  mo.  19  da.  at  6%. 

19.  $357.50  for  9  yr.  3  mo.  10  da.  at  5%. 

20.  $  613.25  for  3  yr.  2  mo.  5  da.  at  6%. 

21.  $5240. 75  for  5  yr.  21  da.  at  5%. 

22.  $3745  for  4  yr.  2  mo.  at  8%. 

23.  $43.75  for  8  yr.  3  mo.  5  da.  at  5%. 

24.  $  745.27  for  6  yr.  9  mo.  18  da.  at  6%. 

25.  $319.50  for  8  yr.  2  mo.  5  da.  at  7%. 

26.  $3246.98  for  1  yr.  6  mo.  15  da.  at  5%. 

27.  $4921.50  for  4  yr.  9  mo.  24  da.  at  7%. 


272 


INTEREST. 


COMPOUND  INTEREST  TABLE. 

Showing  the  amount  of  $  1,  at  various  rates,  compound  int.  from  1  to  20  yr. 


Yrs. 

2£  per  cent. 

3  per  cent. 

3  J  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

1 

1.025000 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159693 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

7 

1.188686 

1.229874 

1.272279 

1.315932 

1.407100 

1.503630 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

12 

1.344889 

1.425761 

1.511069 

1.601032 

1.795856 

2.012197 

13 

1.378511 

1.468534 

1.563956 

1.665074 

1.885649 

2.132928 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.979932 

2.260904 

15 

1.448298 

1.557967 

1.675349 

1.800944 

2.078928 

2.396558 

16 

1.484506 

1.604706 

1.733986 

1.872981 

2.182875 

2.540352 

17 

1.521618 

1.652848 

1.794676 

1.947901 

2.292018 

2.692773 

18 

1.559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

19 

1.598650 

1.753506 

1.922501 

2.106849 

2.526950 

3.025600 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

Yrs. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  cent. 

11  per  cent. 

12  per  cept. 

1 

1.070000 

1.080000 

1.090000 

1.100000 

1.110000 

1.120000 

2 

1.144900 

1.166400 

1.188100 

1.210000 

1.232100 

1.254400 

3 

1.225043 

1.259712 

1.295029 

1.331000 

1.367631 

1.404908 

4 

1.310796 

1.360489 

1.411582 

1.464100 

1.518070 

1.573519 

5 

1.402552 

1.469328 

1.538624 

1.610510 

1.685058 

1.762342 

6 

1.500730 

1.586874 

1:677100 

1.771561 

1.870414 

1.973822 

7 

1.605781 

1.713824 

1.828039 

1.948717 

2.076160 

2.210681 

8 

1.718186 

1.850930 

1.992563 

2.143589 

2.304537 

2.475963 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.558036 

2.773078 

10 

1.967151 

2.158925 

2.367364 

2.593742 

2.839420 

3.105848 

11 

2.104852 

2.331639 

2.580426 

2.853117 

3.151757 

3.478549 

12 

2.252192 

2.518170 

2.812665 

3.138428 

3.498450 

3.895975 

13 

2.409845 

2.719624 

3.065805 

3.452271 

3.883279 

4.363492 

14 

2.578534 

2.937194 

3.341727 

3.797498 

4.310440 

4.887111 

15 

2.759031 

3.172169 

3.642482 

4.177248 

4.784588 

5.473565 

16 

2.952164 

3.425943 

3.970306 

4.594973 

5.310893 

6.130392 

17 

3.158815 

3.700018 

4.327633 

5.054470 

5.895091 

6.866040 

18 

3.379932 

3.996019 

4.717120 

6.559917 

6.543551 

7.689964 

19 

3.616527 

4.315701 

5.141661 

6.115909 

7.263342 

8.612760 

20 

3.869684 

4.660957 

5.604411 

6.727500 

8.062309 

9.646291 

PROMISSORY  NOTES.  273 

PROMISSORY  NOTES. 

346.  A  written  promise  to  pay  a  sum  of  money  at  a 
specified  time  is  called  a  Promissory  Note  or  a  Note. 

347.  The  person  who  signs  the  note  is  the  Maker  or 
Drawer.     The  person  to  whom  it  is  payable  is  the  Payee. 
The  person  who  owns  the  note  is  the  Holder. 

348.  The  sum  promised  to  be  paid  is  the  Face  of  the  note. 

349.  A  higher  rate  of  interest  than  that  authorized  by 
law  is  called  Usury. 

The  penalty  for  making  usurious  contracts  varies  in  the  different 
States,  from  the  loss  of  the  whole  debt  and  interest  to  nothing. 

350.  When  no  rate  of  interest  is  specified,  the  legal  rate 
in  the  place  where  the  note  is  made  is  always  understood. 

SOME   FORMS  OF  NOTES. 
$827.36.  BUFFALO,  N.Y.,  June  1,  1892. 

Four  months  after  date,  I  promise  to  pay  Henry  B.  Samp- 
son, or  order,  Eight  Hundred  Twenty-Seven  -f-£$  Dollars,  for 
value  received.  DAVID  g  GRAHAM 

$3000.  CHICAGO,  ILL.,  July  10,  1892. 

For  value  received,  two  months  after  date,  I  promise  to  pay 
George  D.  Holmes,  or  bearer,  Three  Thousand  Dollars,  with 

interest  SAMUEL  K.  GOODRICH. 

351.  A  note  should  contain  the  place  where  it  was  made, 
the  date,  the  time  when  payable,  the  amount  or  face  of  the 
note  written  in  words,  the  words  "  for  value  received  "  and 
"  with  interest,"  if  such  is  the  contract,  and  the  place  where 
it  is  to  be  paid. 

1.  Notes  are  often  made  payable  at  a  certain  time  after  date.  They 
may  b«  also  made  payable  on  demand,  or  at  a  specified  date,  as  "On 
August  5,  1892,  I  promise  to  pay,"  etc. 

STAND.    AR.  —  18 


274  INTEREST. 

2.  The  amount  of  the  note  should  be  written  in  words. 

3.  If  the  words  "for  value  received  "  are  omitted,  the  owner  of  the 
note  may  have  to  prove  that  the  maker  received  the  value  specified  in 
the  note.     If  the  words  "with  interest"  are  omitted,  the  note  will  not 
draw  interest  until  it  is  due.     After  it  is  due  it  will  draw  interest  at  the 
legal  rate  prevailing  at  the  place  where  it  was  made. 

4.  If  no  place  of  payment  is  named  in  the  note,  it  is  payable  at  the 
maker's  place  of  business. 

352.  A  person  who  writes  his  name  across  the  back  of 
a  note  to  transfer  it  to  another  person  or  to  guarantee  its 
payment,  is  called  an  Indorser. 

The  payee  may  indorse  a  note  by  writing  his  name  and  nothing  else 
on  the  back  of  the  note.  It  is  then  payable  to  the  person  owning  it, 
or  to  the  bearer.  This  is  called  indorsement  in  blank. 

He  may  write  "  Pay  to  A B ."  It  is  then  payable  to  A 

B only. 

He  may  write  "  Pay  to  A B or  order,"  and  it  is  then  pay- 
able to  any  person  to  whom  A B may  order  it  paid.  This  is 

called  special  indorsement. 

He  may  write  "  Pay  A B or  bearer,"  and  it  is  then  payable 

to  the  one  who  presents  it,  or  the  bearer. 

There  are  also  other  forms  of  indorsement. 

353.  A  note  that  is  payable  to  the  order  of  the  payee  or 
to  the  bearer  is  a  Negotiable  Note. 

Thus,  both  the  foregoing  notes  are  negotiable. 

354.  A  note  that  is  payable  to  the  payee  only  is  a  Non- 
Hegotiable  Note. 

Thus,  if  the  words  "  or  order  "  and  "  or  bearer  "  were  omitted  from 
the  above  notes,  the  notes  would  be  non-negotiable. 

355.  A  note  is  payable,  or  is  said  to  mature,  at  the  time 
specified  in  it,  except  in  some  states  where  three  days  extra, 
•called  days  of  grace,  are  allowed  before  payment  can  be 
legally  enforced.     In  these  states  a  note  matures  on  the  last 
day  of  grace. 

1.  Days  of  grace  are  not  allowed  in  Ariz.,  Cal.,  Colo.,  Conn.,  D.  C.,  Del.,  Fla..  Ga., 
Ida.,  111.,  Ind.,  la.,  Kan.,  Ky.,  La.,  Me.,  Md.,  Mass.,  Mich.,  Minn.,  Mo.,  Mont.,  Neb., 
-JS.  H.,  N.  J.,  N.  Y.,  N.  D.,  Ore.,  O.,  Pa.,  R.  I.,  Tenn.,  TJt.,  Va.,  Vt.,  Wash.,  W.  Va., 
Wis.,  Wy. 


PROMISSORY  NOTES.  275 

2.  If,  when  a  note  is  unpaid  at  its  maturity,  the  holder  fails  to 
protest  it,  that  is,  to  notify  the  indorsers  in  a  manner  prescribed  by 
law  that  it  is  unpaid,  they  are  released  from  responsibility  regarding 
its  payment. 

3.  The  protest  must  be  served  upon  the  indorsers  at  the  latest,  upon 
the  day  following  that  upon  which  the  note  matures. 

356.  A  note  signed  by  two  or  more  persons,  who  become 
jointly  and  individually  responsible   for   its   payment,  is 
called  a  Joint  and  Several  Note. 

WRITTEN   EXERCISES. 

357.  1.    Write  a  negotiable  note  for  $500.25,  making 
yourself  the  payee,  and  James  J.  Eogers  the  maker.    Inter- 
est at  the  legal  rate. 

2.  Write  a  non-negotiable  note  for  $315.17,  making  W. 
R.  Howard  the  payee,  payable  on  demand  without  interest. 

3.  Write  two  forms  of  negotiable  notes  for  $  3184.25, 
due  in  three  months  to  James  P.  Hermann,  with  interest. 

4.  Indorse  them  properly  for  transferring  one  to  bearer, 
and  the  other  to  H.  H.  Hurd,  or  order. 

5 .  Write  a  note  from  the  following  data :  face,  $  5000 ; 
negotiable ;  maker,  P.  G.  Sloane ;  payee,  J.  S.  Orton ;  pay- 
able on  demand ;  rate  of  interest,  the  legal  rate. 

6.  Write  a  negotiable  note  for  $1200,  making  David  E. 
Swan  the  payee,  Stephen  Baird  the  maker,  and  payable  on 
demand  with  interest  at  5%. 

7.  Write  a  negotiable  note  for  $  350.75,  payable  to  D.  C. 
Morrison,  due  in  60  days  from  date  with  interest,  and  signed 
by  H.  G.  Goodspeed  &  Co. 

Find  the  interest  of : 

8.  $175  from  Jan.     2,  1877,  to  Oct.    14,1878,    at  6%. 

9.  $380  from  Mar.  14,  1879,  to  Aug.  20,  1880,   at  5%. 

10.  $575  from  Sept.    6,  1880,  to  Oct.      4,  1881,   at  7%. 

11.  $860  from  Mar.  15,  1882,  to  May  31,  1883,   at  5%. 


276  INTEREST. 

PARTIAL   PAYMENTS. 

358.  A  payment  in  part  of  a  note  or  other  obligation  is 
a  Partial  Payment. 

The  payments,  with  the  date  at  which  they  were  paid,  are  usually 
indorsed  upon  the  back  of  the  note  or  other  obligation. 

Business  men  often  settle  notes  and  accounts  running 
for  a  year  or  less,  upon  which,  partial  payments  have  been 
made,  by  the  MERCANTILE  RULE. 

RULE.  —  Find  the  amount  of  the  principal  at  the  time  of 
settlement. 

Find  the  amount  of  each  payment,  from  the  time  it  was 
made  until  the  time  of  settlement,  and  from  the  amount  of  the 
principal  subtract  the  amount  of  the  payments. 

WRITTEN    EXERCISES. 

359.  1.   A  note  of  $760,  dated  Jan.  10,  1890,  was  in- 
dorsed as  follows:   Mar.  13,  1890,  $175;  July  28,  189P, 
$360.     What  remained  due  Dec.  22,  1890,  at  6%  ? 

2.  On  a  note  for  $1245,  dated  Jan.  12,  1890,  were  the 
following  indorsements:    May  15,  1890,  $236;    June   20, 

1890,  $350;    Aug.  10,  1890,  $180;'  Sept.  3,  1890,  $220. 
How  much  was  due  Oct.  30,  1890,  at  6%  ? 

3.  On  a  note  dated  Aug.  15,  1885,  for  $3500,  were  the 
following  indorsements:  Oct.  10,  1885,  $320;  Feb.  5,  1886, 
$476;  Apr.  20,  1886,  $525;.  June  24,  1886,  $700.     What 
amount  was  due  Aug.  3, 1886,  at  1%  ? 

4.  What  is  the  balance  due  Apr.  1,  1892,  on  a  note  for 
$1500,  dated  Apr.  1,  1891,  with  interest  at  8%,  on  which 
the  following  payments  have  been  made:  June  10,  1891, 
$270;  Aug.  23,  1891,  $328;  Sept.  10, 1891,  $145;  Nov.  1, 

1891,  $195;  Feb.  13,  1892,  $200? 


PARTIAL  PAYMENTS.  277 

360.  Most  of  the  States  have  adopted  the  United  States 
Rule  for  computing  the  amount  due,  when  partial  payments 
have  been  made. 

361.  Some  additional  rules  that  have  been  adopted  for 
computing  the   indebtedness   upon   a  promissory   note   or 
other  obligation  will  be  found  under  Art.  614. 

1.  A  note  was  given,  Jan.  1, 1890,  for  $  700.  The  follow- 
ing payments  were  indorsed  upon  it :  May  6, 1890,  $  85  ;  July 
1,  1891,  $40;  Aug.  20,  1891,  $100;  Jan.  10,  1893,  f  350. 
How  much  was  due  Sept.  30,  1894,  with  interest  at  6%  ? 

PROCESS. 

Principal $700.00 

Int.  to  May  6,  1890,— 4  mo.  5  da 14.58 

Amount 714.58 

First  payment 85.00 

New  principal 629.58 

Int.  from  May  6,  1890,  to  July  1, 1891,—  1  yr.  1  mo. 

25  da 43.55 

Second  payment,  less  than  interest  due      .     .     .     .     $40.00 
Int.  on  $629. 58  from  July  1, 1891,  to  Aug.  20, 1891,— 

1  mo.  19  da .  5.14 

Amount 678.27 

Third  payment  to  be  added  to  second $100.00  140.00 

New  principal 538.27 

Int.  from  Aug.  20,  1891,  to  Jan.  10,  1893,  —  1  yr.  4 

mo.  20  da 44.85 

Amount 583.12 

Fourth  payment • 350.00 

New  principal 233.12 

Int.  from  Jan.  10, 1893,  to  Sept.  30, 1894,  —  1  yr.  8  mo.  20  da.  24.08 

Amount  due,  Sept.  30,  1894 $257.20 

UNITED  STATES  RULE.  —  Find  the  amount  of  the  principal 
to  a  time  when  a  payment,  or  the  sum  of  the  payments,  equals 
or  exceeds  the  interest  due,  and  from  this  amount  subtract  such 
payment  or  payments.  With  the  remainder  as  a  new  prin&ipal, 
proceed  as  before. 


278  INTEREST. 

2.  A  note  for  $850,  dated  June  24,  1887,  was  indorsed 
as  follows :  Apr.  1, 1888,  $  250 ;  Nov.  18, 1888,  $  300.     How 
much  was  due  on  the  note  Jan.  1,  1889,  the  rate  of  interest 
being  6%  ? 

3.  A  note  of  $  1000,  dated  Apr.  2,  1881,  was  indorsed  as 
follows:  June  1,  1881,  $200;  Sept.  10,  1881,  $350.     How 
much  was  due  Apr.  2,  1882,  interest  being  at  1%  ? 

4.  A  note  of  $1115,  dated  July  6,  1883,  was  indorsed 
as  follows:  Sept.  15,  1883,  $180;  Jan.  2, 1884,  $225;  Mar. 
20,  1884,  $300.     What  was  due  May  1,  1884,  the  rate  of 
interest  being  6%  ? 

5.  A  note  was  given  Jan.  1,  1885,  for  $750.     The  fol- 
lowing payments  were  indorsed  upon  it :  Mar.  1, 1885,  $125 ; 
July  6,  1885,  $  325 ;  Dec.  10,  1885,  $  75.     How  much  was 
due  June  1,  1886,  with  interest  at  6%  ? 

6.  A  note  of  $900,  dated  Apr.  2,  1886,  was  indorsed  as 
follows:  Sept.  8,  1886,  $115;  June  20,  1887,  $175;  Dec. 
14,  1887,  $200.     What  amount  was  due  May  26,  1888,  with 
interest  at  7%  ? 

7.  A  note  of  $1200,  dated  Jan.  1,  1887,  had  the  follow- 
ing indorsements:  Aug.  1,  1887,  $175;  Dec.  1,  1887,  $225; 
July  1, 1888,  $  250;  Nov.  1, 1888,  $  100.     What  amount  was 
due  Jan.  1,  1889,  with  interest  at  1%  ? 

8.  A  note  of  $1450,  dated  Sept.  20,  1886,  was  indorsed 
as  follows:  Jan.  8,  1887,  $20;  June  8,  1887,  $180;  Oct. 
20, 1887,  $210;  Apr.  15, 1888,  $15.    What  was  due  June  24, 

1888,  with  interest  at  8%  ? 

9.  A  note  of  $1800  dated  Aug.  2,  1887,  was  indorsed 
'as  follows  :  Jan.  4, 1888,  $  200 ;  Jan.  4, 1889,  $  100 ;  June  5, 

1889,  $  500.     What  amount  was  due  Jan.  3,  1890,  with  in- 
terest  at  8%  ? 


PROBLEMS  ifr  SIMPLE   INTEREST.  279 

PROBLEMS  IN  SIMPLE  INTEREST. 

362.  The  principal,  time,  and  interest  given,  to  find  the 
rate. 

1.  What  is  the  interest  of  $  100  for  1  yr.  at  1%  ?     At 
4%  ?     At  6%  ? 

What  was  the  rate : 

2.  When  the  interest  of  $  100  for  1  yr.  was  $  7  ? 

3.  When  the  interest  of  $  100  for  2  yr.  was  $  16  ? 

4.  When  the  interest  of  $200  for  3  yr.  was  $  36  ? 

5.  When  the  interest  of  $  50  for  3  yr.  was  $  15  ? 

WRITTEN   EXERCISES. 

363.  What  is  the  rate  per  cent,  when  the  interest : 

1.  Of  $  450  for  2  yr.  4  mo.  is  $  52.50  ? 

SOLUTION.  —  The  interest  of  $450  for  2  yr.  4  mo.  at  1  %  is  $  10.50. 
$  52.50  --  $  10.50  =  5.      .•.  The  rate  is  5  %. 

2.  Of  $325  for  1  yr.  6  mo.  is  $  19.50  ? 

3.  Of  $  480  for  2  yr.  3  mo.  is  $  64.80  ? 

4.  Of  $  240  for  1  yr.  9  mo.  is  $  29.40  ? 

5.  Of  $ 375  for  1  yr.  5  mo.  is  $ 31.87$  ? 

6.  Of  $  500  for  2  yr.  2  mo.  is  $  26.25  ? 

7.  Of  $  475  for  3  yr.  4  mo.  is  $  95  ? 

8.  A  house  that  cost  $6000  was  rented  for  $490.     If 
$  100  was  paid  for  taxes  and  repairs,  what  rate  of  interest 
did  the  purchase  money  yield  ? 

9.  Mr.  Donat  borrowed  $  1575  on  ther  1st  of  April.     On 
the  1st  of  November  following  he  paid  the  amount,  which. 
was  $1630.125.    What  rate  of  interest  did  he  pay  ? 

10.  The  annual  income  of  a  farm  that  cost  $10,500  was- 
$  785.  The  expenses  were  $  260.  What  rate  per  cent  did 
the  farmer  realize  on  his  investment  ? 


280  INTEREST 

364.  The  principal,  rate,  and  interest  given,  to  find  the  time. 

1.  How  much  is  the  interest  of  $100  for  1  yr.  at  6%  ? 

2.  How  much  is  the  interest  of  $300  for  1  yr.  at  5%  ? 
For  5  yr.  ? 

3.  $100  loaned  at  6%,  brings  an  income  of  $24.     For 
how  long  was  it  loaned  ?     How  long,  when  the  interest  was 

$18?     $24?     $3?     $4?     $2?     $1.50? 

WRITTEN    EXERCISES. 

365.  In  what  time  will : 

1.  $  280  produce  $  25.20,  with  interest  at  6%  ? 
SOLUTION.  —The  interest  of  $280  for  1  yr.  at  6 %  is  $  16.80. 

$25.20  -*-  $16.80  =  1|.          .-.  The  time  is  l|yr. 

2.  $  300  produce  $  37.50  interest  at  5%  ? 

3.  $480  produce  $74.28  interest  at  3%  ? 

4.  $  400  produce  $  62.06|  interest  at  7%  ? 

5.  $940  produce  $432.40  interest  at  6%  ? 

6.  $  860  produce  $  247.25  interest  at '5%  ? 

7.  $  984  produce  $  288.64  interest  at  8%  ? 

8.  $998  produce  $185.145  interest  at  5%  ? 

9.  $  1200  produce  $  1200  interest  at  7%  ? 

10.  $  1500  produce  $1500  interest  at  8%  ? 

11.  How  long  must  $530  be  at  interest  at  6%  to  amount 
to  $  641.30  ? 

12.  A  man  borrowed  $  1200  at  5-|-%,  and  retained  it  until 
it  was  doubled.     How  long  did  he  have  it  ? 

13.  When  will  $475,  put  at  interest  at  6%  April  1, 1891, 
amount  to  $489.25. 

14.  A  certain  sum  of  money  was  put  at  interest  at  8% 
June  24, 1850.    When  was  the  interest  double  the  principal  ? 


PROBLEMS  IN   SIMPLE  INTEREST.  281 

366.  The  rate,  time,  and  interest  given,  to  find  the  principal. 

1.  At  6%  what  sum  will  yield  an  income  annually  of 
$6?     Off  12?     Off  18?     Off  3?     Of$4?     Off2? 

2.  At  6%  what  sum  will  yield  an  income  of  $12  in 
2  yr.?     Of  .f  18  in  3  yr.  ?     Of  f  15  in  1  yr.  ?     Of  $25  in 
5yr.?     Of  $4inlyr.? 

WRITTEN    EXERCISES. 

367.  1.  What  sum  of  money  at  4J-%  will  produce  $  75.40 
interest  in  3  yr.  4  mo.  ? 

Int.  of  $  1  for  3  yr.  4  mo.  at  4£  %  =  $  .15. 

$75.40  H-  $.15  =  502.66|,  .-.  $502.66f  is  the  principal. 

EXPLANATION.  —  Since  the  interest  of  $  1  for  3  yr.  4  mo.,  at  4|  %,  is 
$  .15,  it  will  require  a  sum  equal  to  as  many  times  $  1  to  produce  $75.40 
interest,  as  $  .15  is  contained  times  in  that  number,  or  $502.66f. 

What  sum  of  money  will  produce : 

2.  $  25.50  interest  in  2  yr.  at  5%  ? 

3.  $  33.75  interest  in  2  yr.  3  mo.  at  6%  ? 

4.  $  43.86  interest  in  3  yr.  4  mo.  at  6%  ? 

5.  $49.75  interest  in  6  mo.  18  da.  at  7%  ? 

6.  $  50.32  interest  in  5  mo.  27  da.  at  8%  ? 

7.  $38.40  interest  in  9  mo.  15  da.  at  9%  ? 

8.  $  45.80  interest  in  2  mo.  21  da.  at  6%  ? 

9.  $  68.50  interest  in  7  mo.  25  da.  at  5%  ? 

10.  $  95.35  interest  in  4  yr.  7  mo.  at  7%  ? 

11.  What  principal  at  6%,  loaned  from  June  24, 1891,  to 
Sept.  10,  1893,  will  amount  to  $  2575  ? 

12.  What  principal  will  produce  $17.78  interest  from 
Jan.  10,  1892,  to  March  13,  1892,  at  6%  ? 

13.  What  may  I  offer  for  a  residence  which  pays  $  895 
rent  per  year,  so  that  I  may  receive  §\%  interest  on  the 
investment  ? 


282  INTEREST. 


TRUE    DISCOUNT. 

368.  1.    What  will  be  the  amount  of  $100  in  1  yr.  at 
6%  ?     In  2  yr.  ? 

2.  What  is  the  value  now  of  $106  to  be  paid  in  1  yr., 
when  money  is  loaned  at  6%  ?     Of  $112  to  be  paid  in 
2  yr.  ?     Of  $  118  to  be  paid  in  3  yr.  ? 

3.  What  is  the  present  value  of  $212  to  be  paid  in  1  yr., 
when  money  is  loaned  at  6%  ?     Of  $224  to  be  paid  in 
2yr.? 

4.  What  is  the  present  worth  of  a  debt  of  $  672  due  in  1-J- 
yr.,  when  money  is  loaned  at  8%  ?     Of  $  348  due  in  2  yr.  ? 

369.  A  deduction  made  from  a  debt  is  termed  a  Discount. 

370.  A  sum  of  money  which,  when  put  at  interest  at  a 
specified  rate,  will  amount  to  a  debt  when  it  becomes  due, 
is  the  Present  Worth  of  the  debt  due  at  some  future  time. 

371.  The  difference  between  a  debt  and  its  present  worth 
is  the  True  Discount. 

WRITTEN    EXERCISES. 

372.  1.    What  is  the  present  worth  of  a  debt  of  $975.50 
payable  in  1  yr.^6  mo.,  when  money  is  loaned  at  6%  ?    What 

the  discount  ? 

$  1.09  =  Amount  of  $  1  for  H  yr. 

$975.50  -T-  $1.09  =  894.95,  .-.  $894.95  is  the  present  worth. 
$975.50  -  $894.95  =  $80.55,  True  discount. 

EXPLANATION.  —  Since  every  dollar  put  at  interest  now  at  6  %  will 
amount  to  $1.09  in  1  yr.  6  mo.,  it  will  require  as  many  dollars  now 
to  amount  to  $  975.50  as  $  1.09  is  contained  times  in  $  975.50,  or  $  894.95. 

The  debt  $975.50  —  $894.95  =  $  80.55,  the  true  discount. 

EULE. — Divide  the  amount  due,  by  the  amount  of  $1,  for 
the  given  time  and  rate,  and  the  quotient  will  be  the  present 
worth. 


TRUE   DISCOUNT.  283 

Subtract  the  present  worth  from  the  amount  due}  and  the 
remainder  will  be  the  true  discount. 

What  are  the  present  worth  and  discount  of  the  follow- 
ing: 

2.  $  576.75  payable  in  9  mo.,  when  money  is  worth  6%  ? 

3.  $  760.85  payable  in  10  mo.,  when  money  is  worth  5%  ? 

4.  $  437.50  payable  in  1^  yr.,  when  money  is  worth  7%  ? 

5.  $  648.60  payable  in  2  yr.,  when  money  is  worth  5^%  ? 

6.  $1200  payable  in  1  yr.  4  mo.  18  da.,  when  money  is 
worth  6%  ?     When  money  is  worth  5%  ? 

7.  $1608  payable  in  1  yr.  3  mo.  20  da.,  when  money  is 
worth  S%  ?     When  money  is  worth  6%  ? 

8.  $2575  payable  in  5  mo.  17  da.,  when  money  is  worth 
l\c/0  ?     When  money  is  worth  6|%  ? 

9.  $  1357.85  payable  in  90  da.,  when  money  is  worth 
8%  ?     When  money  is  worth  1\  %  ? 

10.  $3180.50  payable  in  2  yr.  3  mo.  21  da.,  when  money 
is  worth  ¥>\%  ? 

11.  A  owes  $175.90,  due  in  3  yr.  8  mo.,  which  he  wishes 
to  pay  immediately.      How  much  should  he   pay,   money 
being  worth  5%  ? 

12.  A  merchant  was  offered  a  credit  of  3  months  on  a 
bill  of  goods  amounting  to  $3468,  or  a  discount  of  2%  for 
cash.     How  much  better  was  the  latter  offer,  money  being 
worth  7%  ? 

13.  Mr.  Hyatt  owes  me  $460.75,  due  in  8  mo.  15  da.     If 
he  desires  to  pay  me  now,  what  sum  should  I  accept,  money 
being  worth  §\°/o  ? 

14.  What  is  the  difference  between  the  true  discount  of 
$248.76,  due  in  2  yr.  3  mo.  15  da.,  and  the  interest  of 
$248.76  for  2  yr.  3  mo.  15  da.,  money  being  worth  6%  ? 


284  INTEREST. 

BANK  DISCOUNT. 

373.  An  institution  chartered  under  the  law  to  receive 
money  for  safe  keeping,  to  loan  money,  or  to  issue  notes  or 
bills  to  circulate  as  money,  is  called  a  Bank. 

374.  A  considerable  part  of  the  business  of  most  banks  is 
the  paying  of  notes  before  they  are  due. 

375.  If  a  bank  becomes  satisfied  that  a  note  is  valid  or 
properly  secured  by  indorsement,  it  may  advance  the  sum 
due  at  maturity  (Art.  355)  less  the  simple  interest  on  that 
sum  for  the  time  the  note  has  still  to  run.     The  note,  which 
is  retained  at  the  bank,  is  then  said  to  be  discounted. 

Banks  usually  discount  for  short  periods,  not  exceeding  3  or  4  mo. 

376.  The  number  of  days  from  the  time  a  note  is  dis- 
counted to  the  time  when  it  legally  matures  is  called  the 
Term  of  Discount. 

1.  In  a  majority  of  the  states  and  territories  when  a  note  falls  due 
on  Sunday   or  on  a  legal  holiday,  it  matures  on  the  next  succeeding 
business  day,  and  the  term  of  discount  includes  that  day  ;   conse- 
quently, in  such  of  these  states  as  allow  days  of  grace,  4,  5,  or  even  6 
days  of  grace  may  be  allowed  before  a  note  legally  matures. 

However,  in  a  few  states,  a  note  matures  on  the  business  day  next 
preceding,  but  the  term  of  discount  is  considered  to  be  the  full  time. 

2.  A  list  of  the  states  and  territories  which  do  not  allow  days  of 
grace  is  given  in  Art.  355,  Note  1. 

Throughout  this  book,  in  examples  involving  the  term  of  discount 
of  notes,  when  no  place  is  mentioned,  days  of  grace  are  to  be  reckoned. 

377.  Simple  interest  collected  in  advance  for  the  term  of 
discount  upon  the  sum  due  on  a  note  at  its  maturity  is  called 
Bank  Discount. 

A  bank  usually  demands  that  the  notes  which  it  discounts  be  made 
payable  at  that  bank. 

378.  The  sum  due  on  a  note  at  its  maturity,  less  the  bank 
discount,  is  called  the  Proceeds  or  Avails  of  a  note. 


BANK  DISCOUNT.  285 

379.  To  find  the  time  when  notes  mature,  the  term  of  dis- 
count, the  discount,  and  the  proceeds : 

1.  $  350.86.  BUFFALO,  N.Y.,  July  5,  1899. 
Three  months  after  date,  for  value  received,  I  promise  to 

pay  David  B.  Graham,  or  order,  Three  Hundred  Fifty  ^j- 
Dollars,  at  the  First  National  Bank. 

DANIEL  R.  SLAUSON. 
Discounted  August  1, 1899,  at  6%. 

SOLUTION. 

Since  days  of  grace  are  not  allowed  in  the  State  of  New  York,  the 
note  matures  on  Oct.  5. 

The  term  of  discount  is  from  Aug.  1  to  Oct.  5,  65  da. 

The  bank  discount  is  the  interest  of  $  350.86  at  6  %  for  65  da.,  which 
is  $3.80. 

The  proceeds  =  $  350.86  -  $  3.80  =  $  347.06. 

NOTE.  —  Although  the  time  in  the  note  is  expressed  by  months,  the 
term  of  discount  is  reckoned  by  counting  the  actual  number  of  days 
from  the  date  of  discount  to  the  date  of  maturity. 

2.  $  685.30.  MONTGOMERY,  ALA.,  Jam.  7,  1892. 
Sixty  days  after  date,  I  promise  to  pay  to  the  order  of 

William  S.  Watson,  Six  Hundred  Eighty-five  ^  Dollars, 
for  value  received,  at  The  Commercial  National  Bank. 

HENKY  G.  DANFORTH. 
Discounted  Feb.  3,  1892,  at  8%. 

SOLUTION. 

The  note  matures  nominally  60  days  after  Jan.  7,  or  on  Mar.  7  ;  but 
since  3  days  of  grace  are  allowed  in  Alabama,  the  note  legally  ma- 
tures on  Mar.  10.  The  date  of  maturity  is  commonly  indicated  thus  : 
Mar.  Vio- 

The  term  of  discount  is  from  Feb.  3  to  Mar.  10,  36  da. 

The  bank  discount  is  the  interest  of  $  685.30,  at  8  % ,  for  86  da., 
counting  360  da.  a  year. 

The  discount  is,  therefore,  $  5.48. 

The  proceeds  =  $ 685.30  -  $  5.48  =  $ 679.82. 

NOTE.  —  In  computing  interest  by  days,  360  days  are  usually  con- 
sidered a  year. 


286  INTEREST. 

3.  $  764.75.  PORTLAND,  ME.,  Feb.  4,  1896. 

Three  months  after  date,  for  value  received,  I  promise  to 
pay  Francis  Damon,  or  order,  Seven  Hundred  Sixty-four  Jfa 
Dollars,  at  the  Girard  Bank. 

D.  B.  BARTON. 

Discounted  Mar.  10,  1896,  at  6  %. 

4.  $537.45.  PROVIDENCE,  R.L,  May  14,  1896. 

Three  months  after  date,  for  value  received,  I  promise  to 
pay  Henry  E-.  Grover,  or  order,  Five  Hundred  Thirty-seven 
•3^  Dollars,  at  the  First  National  Bank. 

DONALD  MCNAUGHTON. 
Discounted  May  25,  1896,  at  6  %. 

5.  $850.50.,  RICHMOND,  VA.,  Oct.  6,  1899. 

Sixty  days  after  date,  for  value  received,  I  promise  to  pay 
William  Sanf ord,  or  order,  Eight  Hundred  Fifty  -ffa  Dollars, 
at  the  Union  Bank. 

SAMUEL  J.  DUNDSON. 

Discounted  Nov.  1,  1899,  at  6  %. 

6.  $235.68.  RICHMOND,  VA.,  JAN.  8,  1896. 

Four  months  after  date,  for  value  received,  I  promise  to 
pay  C.  F.  Cramer,  or  order,  Two  Hundred  Thirty-five  -ffa 
Dollars,  at  the  Merchants'  Bank. 

HENRY  C.  PEAKE. 

Discounted  April  12,  1896,  at  6  %. 

7.  $472.48.  SAN  FRANCISCO,  CAL.,  April  2,  1898. 

Five  months  after  date,  for  value  received,  I  promise  to 
pay  R.  B.  Goodrich,  or  order,  Four  Hundred  Seventy-two  ^nr 
Dollars,  at  the  Citizens'  Bank. 

H.  BOYLE  THOMPSON. 

Discounted  May  29, 1898,  at  7  %. 


BANK  DISCOUNT.  287 

8.  $1000.  RALEIGH,  N.C.,  June  24,  1891. 
Four  months  after  date,  for  value  received,  I  promise  to 

pay  Mary  C.  Platt,  or  order,  One  Thousand  Dollars,  at  the 
First  National  Bank. 

DRAPER  S.  ANDREWS. 
Discounted  Sept.  10,  1891,  at  5%. 

9.  $1100.  LITTLE  ROCK,  ARK.,  July  5,  1891. 
Three  months  after  date,  I  promise  to  pay  G.  E.  Fillmore, 

or  order,  Eleven  Hundred  Dollars,  at  the  Mechanics'  Bank, 
with  interest  at  6  %.     Value  received. 

J.    T.    HOSMER. 

Discounted  Aug.  5,  1891,  at  6  %. 

If  a  note  bears  interest,  find  the  discount  on  the  amount  of  the  note 
at  its  maturity. 

10.  $  135T%.  JACKSON,  Miss.,  May  3,  1891. 
Sixty  days  after  date,  I  promise  to  pay  F.  H.  Stowell,  or 

order,  One  Hundred  Thirty-five  -££$  Dollars,  with  interest 
at  6  %,  for  value  received. 

A.  L.  MUNSON. 
Discounted  May  2Q,  1891,  at  6  %. 

11.  $637.85.  CARSON  CITY,  NEV.,  Feb.  16,  1891. 
Four  months  after  date,  for  value  received,  I  promise  to 

pay  to  the  order  of  C.  G.  Lamson,  Six  Hundred  Thirty- 
seven  ^5-  Dollars,  with  interest  at  7%. 

SPENCER  C.  GRANGER. 
Discounted  Apr.  4,  1891,  at  7%. 

12.  $1200.  GALVESTON,  TEX.,  Aug.  3,  1891. 

Ninety  days  after  date,  I  promise  to  pay  Peter  K.  Good- 
win, or  order,  Twelve  Hundred  Dollars,  for  value  received, 
at  the  First  National  Bank. 

EGBERT  C.  CROPSEY. 

Discounted  Sept.  2,  1891,  at  8%. 


288  INTEREST. 

380.  To  find  the  face  of  a  note  when  the  proceeds,  time,  and 
rate  are  given. 

1.  What  is  the  bank  discount  of  a  note  for  $  1,  due  in 
1  mo.  27  da.  at  6%?     What  are  the  proceeds  ? 

2.  Since  $.99  is  the  proceeds  of  $  1  when  discounted  at 
a  bank  for  1  mo.  27  da.  at  6%,  of  what  sum  is  $  1.98  the 
proceeds  for  the  same  time  and  rate?     $2.97?     $3.96? 
$4.95? 

3.  Of  what  sum  is  $  1.97  the  proceeds  when  a  note  is 
discounted  at  a  bank  for  2  mo.  27  da.  ?     $  2.955  ? 

WRITTEN   EXERCISES. 

381.  1.    The   proceeds  of  a  note  for  2  mo.  12   da.  dis- 
counted at  a  bank  at  7%  were  $  1182.50.     What  was  the 
face  of  the  note? 

SOLUTION. 


$  1  -  $  .0145|  =  $  .9854£,  Proceeds  of  $  1. 
$  1182.50  -r-  $  .9854£  =  1200,  .-.  $  1200  is  the  face  of  note. 

KULE.  —  Divide  the  proceeds  by  the  proceeds  of  $  1  at  the 
given  rate  for  3  days  more  than  the  specified  time. 

2.  The  proceeds  of  a  note  for  3  mo.  when  discounted  at 
a  bank  at  6%  were  $  590.70.     What  was  its  face  ? 

3.  What  must  be  the  face  of  a  note  at  60  da.,  the  proceeds 
of  which  when  discounted  at  a  bank  at  6%,  are  $  336.43? 

4.  The   proceeds  of  a  note  for  1  mo.  18  da.  when  dis- 
counted at  a  bank  at  5%  were  $  1869.35.     What  was  its 
face? 

5.  A   gentleman  wishes  to  raise  $  1000  by  having  his 
note  for  2  mo.  discounted  at  a  bank  at  6%.     What  must  be 
the  face  of  the  note  ? 

6.  For  what  sum  must  a  note  for  2  mo.  17  da.  be  made 
so  that  the  proceeds  after  it  has  been  discounted  at  a  bank 
at  7%  may  be  $895? 


STOCKS  AND  BONDS.  289 


STOCKS   AND   BONDS. 

382.     1.    Into  how  many  shares  of  $100  each  can  the 
capital  stock  of  a  company  amounting  to  $  100,000  be  divided. 
Shares  will  be  regarded  as  $  100  each  unless  otherwise  specified. 

2.  How  much  of  the  capital  does  a  man  own  who  has 
50  shares  ?     75  shares  ?     100  shares  ? 

3.  How   much    stock   is   represented   by   a   certificate 
entitling  the  holder  to  50  shares  ?     To  100  shares  ? 

4.  What  is  the  market  value  of  20  shares  of  stock  when 
it  is  sold  at  the  original  or  par  value  ? 

5.  What  is  the  market  value  of  20  shares  of  stock  when 
it  is  sold  at  5%  above  the  original  or  par  value? 

6.  What  is  the  market  value  of  20  shares  of  stock  when 
it  is  sold  at  5%  below  the  original  or  par  value  ? 

7.  What  will  be  the  cost  of  a  share  of  stock  at  5%  above 
par  value,  if  I  pay  a  stockbroker  \%  of  the  par  value  of  the 
stock  for  purchasing  it  ? 

8.  What  will  be  the  cost  of  a  share  of  stock  at  5%  below 
par  value,  if  I  pay  J%  for  purchasing  it  ? 

9.  What  is  the  value  of  10  shares  of  bank  stock  at  90% 
of  its  par  value  ? 

10.  What  will  4  shares  of  stock  cost  at  5%  above  the  par 
value,  if  \°f0  of  the  par  value  is  paid  to  the  broker  for  pur- 
chasing it  ? 

11.  A  company  with  a  capital  stock  of  $100,000  gained 
$  10,000  above  its  expenses.     What  %  of  the  capital  stock 
was  the  gain,  and  if  the  gain  was  divided  among  the  stock- 
holders, what  °/0  of  dividend  did  each  receive  ? 

STAND.   AR.  — 19 


290  INTEREST. 

12.  If  the  gain  or  dividend  upon  the  capital  stock  is  10%, 
when  money  is  loaned  at  5%,  would  the  stock  sell  above  or 
below  par  ? 

13.  If  the  dividend  was  only  1%  of  the  capital  stock, 
how  would  the  stock  sell  ? 

14.  If  a  company  whose  capital  stock  is  $  100,000  loses 
$  10,000,  what  %  of  the  capital  stock  is  the  deficiency  or 
loss  ? 

15.  If  the  loss  must  be  made  good  by  the  stockholders, 
how  much  must  a  man  pay  who  owns  10  shares  of  the  stock, 
or  what  will  be  his  assessment? 

16.  What  will  be  the  annual  income  of  a  written  obliga- 
tion called  a  bond  for  $  10,000,  if  it  pays  10%  interest  annu- 
ally ? 

383.  When  a  large  sum  of  money  is  to  be  raised  for  the 
purpose  of  carrying  on  some  enterprise,  usually  a  number  of 
people  contribute  a  portion  of  the  sum  or  capital  stock,  and 
thus  form  a  Company. 

384.  As  soon  as  the  money  is  subscribed  or  raised,  a 
Charter,  or  legal  document,  which  defines  the  powers  and 
limitations  of  the  company,  is  obtained. 

One  of  the  special  advantages  of  a  charter  is  that  it  commonly  limits 
each  stockholder's  liability  to  the  amount  he  has  contributed,  whereas 
as  a  member  of  an  unincorporated  company  or  a  firm,  each  person  is 
liable  for  all  the  debts  of  the  company. 

385.  Each  person  receives  a  Certificate  which  shows  what 
amount  he  has  contributed.    This  certificate  usually  specifies 
the  number  of  shares  of  stock  to  which  the  person  is  entitled, 
and  the  original  value  of  each  share. 

386.  The  value  ©f  the  shares  of  stock  named  in  the  cer- 
tificate is  called  the  Par  Value. 


STOCKS  AND  BONDS.  291 

When  the  business  is  very  profitable,  the  market  value  of  the  shares 
of  stock  is  high  or  above  par,  or  at  a  premium  ;  when  the  business  is 
unprofitable  the  shares  are  low  or  below  par,  or  at  a  discount. 

387.  When  the  stock  subscribed  by  a  stockholder  is  not 
all  paid  in  at  one  time,  the  several  portions  paid  in  are 
called  Installments. 

388.  When  the  business  of  the  company  has  been  pros- 
perous, the  gain  is  divided  among  the  stockholders,  and  each 
one's  share  is  termed  a  Dividend. 

389.  When  the  business  has  not  been  prosperous,  the 
stockholders  are  required  to  make  up  the  deficiency  by  an 
Assessment. 

390.  The  prices  of  stocks  vary  according  to  the  prosperity 
of  the  business,  so  that  some  stocks  sell  above  par,  and 
others  below.     The  sum  for  which  stocks  sell  is  called  the 
Market  Value. 

391.  A  written  obligation  under  seal,  securing  the  pay- 
ment of  a  sum  of  money  before  a  specified  time,  is  called  a. 
Bond. 

When  the  United  States,  any  state,  city,  county,  town,  village,  or 
incorporated  company  wishes  to  raise  funds  for  some  purpose,  bonds, 
are  prepared  and  sold.  The  bonds  are  thus  secured  by  the  property 
of  those  who  issue  them,  and  bear  a  fixed  rate  of  interest  payable 
annually,  semiannually,  or  quarterly.  , 

392.  When  the  bonds  are  recorded  by  their  numbers  and 
the  names  of  the  persons  owning  them,  they  are  called 
Registered  Bonds. 

Registered  Bonds  cannot  be  transferred  without  a  change  being: 
made  in  the  record  kept  by  the  company. 

393.  Bonds  to  which  interest  certificates,  called  Coupons> 
are  attached  are  called  Coupon  Bonds. 

The  coupons  are  cut  off  and  presented  for  payment,  at  banks  or 
elsewhere,  when  interest  is  due. 


292  INTEREST. 

394.  Government  bonds  and  state  bonds  are  of  various 
kinds,  and  they  are  briefly  described  by  abbreviations  for 
rate  of  interest,  date  of  payment,  etc. 

Thus,  U.S.  4's,  1907,  reg.,  means  United  States  registered  bonds, 
bearing  4%  interest,  payable  in  1907. 

Bonds  are  discussed  with  Stocks  because  they  are  bought  and  sold 
at  the  Stock  Exchange,  though  there  is  little  intrinsically  common  to 
Stocks  and  Bonds.  Bonds  pay  regular  interest  at  fixed  rates;  the 
income  from  Stocks  is  variable. 

395.  A  person  whose  business  it  is  to  buy  and  sell  stocks 
is  called  a  Stock  Broker,  and  the  compensation  he  receives 
for  his  services  is  called  Brokerage. 

396.  PRINCIPLE.  —  Brokerage  is  computed  upon  the  par 
value  of  the  stock. 

WRITTEN    EXERCISES. 

397.  1.    What  is  the  cost  of  500  shares  Hanover  and 
King's  Point  Canal  Co.  stock  at  50^-,  brokerage  %%  ? 


50f  %  of  $  100  =  $  50.62^,  cost  of  1  share. 
$  50.621  x  500  =  $  25312.50,  the  entire  cost.   ' 

EXPLANATION.  —  Since  50^  %  of  the  par  value  of  the  stock  is  the  price 
paid  for  it,  the  entire  cost  of  the  stock,  including  the  rate  for  brokerage, 
is  50f  %  of  the  par  value  of  the  stock.  And  since  the  par  value  of  a 
share  of  the  stock  is  $100,  the  cost  of  a  share  will  be  50f  %  of  $  100,  or 
$50.62^,  and  the  cost  of  500  shares  of  the  stock  will  therefore  be 
500  times  f  50.62  1,  or  $25312.50. 

2.  Find  the  cost   of  150  shares  Canadian  Pacific  R.R. 
stock  at  89,  brokerage  -J-%. 

3.  How  much  will  76  shares  C.,  B.,  &  Q.  R.R.  stock  cost 
at  102,  brokerage  \%  ? 

4.  What  will  be  the  cost  of  45  shares  U.S.  Express  Co. 
stock  at  55,  brokerage  J-%  ? 


STOCKS  AND  BONDS.  293 

5.  How  much  must  be  paid  for  120  shares  Columbia 
Coal  Co.  stock  at  33f,  brokerage  %%  ? 

6.  What  must  be  paid  for  $  6000  in  U.S.  currency,  5's, 
'96  at  8J%  premium,  brokerage  \°/0  ? 

7.  How  much  will  125  shares  N.Y.  C.  &  H.  E.  E.E.  stock 
cost  at  113^,  brokerage  \°/0  ? 

8.  I  bought  180  shares  Long  Island  E.E.  stock  at  94, 
and  afterwards  sold  them  at  101J-.     How  much  did  I  gain, 
the  brokerage  in  each  case  being  ^%  ? 

9.  What  will  be  the  cost  of  85  shares  of  railroad  stock 
at  8J%  discount,  brokerage  \°/0  ? 

10.  What  must  be  paid  for  375  shares  Telegraph  stock  at 
12 J  %  premium,  brokerage  -J-%  ? 

11.  What  will  be  the  cost  of  $12,500  in  U.S.  4's,  '97 
reg.,  at  16f  %  premium,  brokerage  \°/G  ? 

12.  I  bought  130  shares  Eock  Island  E.E.  stock  at  106J, 
and  afterwards  sold  them  at  109J.     What  was  the  gain, 
brokerage  in  each  case  |-%  ? 

13..  How  many  shares  of  bank  stock,  at  5%   discount, 
can  be  purchased  for  $  3805,  if  \°/G  is  paid  for  brokerage  ? 
100%  —  5%  =  95%,         '  EXPLANATION.  —  Since   the   stock  was 
#,  —  Q5i  <#,        bought  at  5%  discount,  it  was  bought  at 

«<^i       /n'      95%  °f  its  par  value'  but  the  broker^ 
-  $  yt>  g-  =  4U.       increased  the  cost  |%,  so  that  each  dol- 
lar's worth  of  stock  cost  95^%  of  its  par 

value,  or  $  95|  per  share.     Therefore,  as  many  shares  of  stock  can  be , 
bought  for  $  3805  as  $  95£  is  contained  times  in  $  3805,  which  is  40 
times.     Therefore  40  shares  can  be  bought. 

14.  How  many  shares  Oregon  Navigation  Co.'s  stock,  at 
78,  can  be  bought  for  $  9375,  brokerage  \%  ? 

15.  How  many  shares  Eeading  E.E.  stock,  at  61^,  can 
be  bought  for  $6874,  brokerage  %%? 

16.  Find  the  number  of  pipe  line  certificates,  at  115f, 
that  can  be  bought  for  $  18,520,  each  certificate  being  $  100. 


294  INTEREST. 

17.  What    income    will    be    realized    from    investing 
$4190.63  in  5%  stock,  purchased  at  93,  allowing  £%  for 
brokerage  ? 

$  4190.63  -*-$  93£  =  45. 
Par  value  45  shares  =  $  4500. 

$  4500  x  .05  =  $  225,  annual  income. 

EXPLANATION.  — Since  the  stock  cost  93 1%  of  its  par  value,  every 
ehare  cost  $93| ;  and  as  many  shares  can  be  bought  for  $4190.63  as 
$  93£  is  contained  times  in  that  sum,  which  is  45  times,  or  45  shares. 
Since  the  stock  paid  5  %  income,  the  entire  income  from  45  shares  or 
$4500  is  5%  of  $4500,  which  is  $225. 

18.  What   will   be   the   annual   income   from   investing 
$3457.50  in  5%  stock,  purchased  at  57-J-,  allowing  \°/G  for 
brokerage  ? 

19.  What  income  will  be  derived  from  $4565  invested 
in  Mich.  7's  at  114,  brokerage  \%  ? 

20.  What  income  will  a  man  derive  from  $10,777.375 
invested  in  railroad  bonds  paying  an  annual  dividend  of 
10%,  if  he  buys  them  at  98f,  brokerage  \%  ? 

21.  Which   is   the  more  profitable,   and  how  much,  to 
invest  $6000  in  6%  stock  at  75,  or  in  5%  stock,  purchased 
at  60%  ? 

22.  How  much  will  be  realized  from  investing  $  15,180 
in  4^%  bonds,  purchased  at  94f,  brokerage  \%  ? 

23.  How  much  must  be  invested  in  6%  stock,  purchased 
at   90,   to   secure  to   the   purchaser   an  income   of    $900 

annually  ? 

$900 --$6  =  150. 

Par  value  150  shares  =  $  15,000. 

$  15,000  x  .90  =  $  13,500,  cost  of  stock. 

EXPLANATION.  —Since  the  income  from  1  share  is  $6,  it  will  re- 
quire as  many  shares  to  secure  an  income  of  $  900  as  $  6  is  contained 
times  in  $900,  which  is  150  times,  or  150  shares  ;  and  since  the  stock 
is  selling  at  90%  of  its  par  value,  90%  of  $15,000,  which  is  $13,500, 
is  the  cost. 


STOCKS  AND  BONDS.  295 

24.  How   much   must   be   invested    in  7%   city  bonds, 
bought  at  101^-,  brokerage  £%,  to  yield  an  annual  income 
of  $  840  ? 

25.  How  much  must  I  invest  in  D.  and  H.  Canal  Co. 
R.R.  stock  at  142,  brokerage  -|-%,  to  secure  an  income  of 
$  1600,  if  the  stock  pays  a  dividend  of  10%  ? 

26.  What  sum  must  be  invested  in  U.  S.  4's  at  121  J, 
brokerage  at  |-%,  to  secure  an  annual  income  of  $900  ? 

27.  When  Wisconsin  Central  5's  are  selling  at  95^-,  how 
much  must  be  invested  to  produce  an  income  of  $1000, 
brokerage  -|-%  ? 

28.  What   per   cent   income   on   my  investment  will  I 
receive,  if  I  buy  6%  stock  at  20%  premium  ? 

EXPLANATION.  —  Since   1   share   of 

a*  /»       ap-ioA        AS-         Krrf       the  stock  costs  $120,  and  the  income 
$  6  -  $  120  =  .05,  or  5%.     from  .fc  .g  $  ^  ^  .;come  .g          Qr 


5%  of  the  investment. 

29.  A  man  received  6%  dividend  on  stock  bought  at  25% 
below  par.     What  rate  of  interest  did  he  receive  on  his 
investment  ? 

30.  What  is  the  rate  per  cent  of  income  realized  from 
6%  bonds  bought  at  90  ? 

31.  How  much  must  I  pay  for  New  York  6's  so  that  I 
may  realize  an  income  of  9%  on  the  investment  ? 

<c  ft  _._  Q9__®ggj.  EXPLANATION.  —  Since  I  wish  to 

/>/>o^/     JT  ^          realize  9%  on  my  investment  and 

$  66}  =  66f  %  of  par  value.    the  stock/0yields  ^  income  of  $6 

per  share,  $  6  must  be  9  %  of  the  price  I  should  pay  for  the  stock. 
Therefore,  I  must  pay  $66f  per  share  or  66  f  %  of  its  par  value. 

32.  What  must  be  paid  for  stock  which  pays  a  dividend 
of  10%,  so  as  to  realize  7%  on  the  investment  ? 

33.  How  much  must  I  pay  for  stock  which  pays  a  divi- 
dend of  12%  so  that  I  may  realize  8%  on  the  investment  ? 


296  INTEREST. 

REVIEW   EXERCISES. 
ORAL  EXERCISES. 

398.    1.  A  man  who  had  $360  spent  25%  of  his  money. 
How  much  had  he  left  ? 

2.  B's  salary  was  $1500  per  year,  and  he  saved  33^% 
of  it.     How  much  did  he  spend  ? 

3.  In  selling  a  suit  of  clothes  a  merchant  took  10%  less 
than  the  price  asked  and  received  $  36.     What  was  the 
asking  price  ? 

4.  If  a  man  who  earns  $60  a  month  spends  $45  for 
necessary  expenses,  what  per  cent  of  his  earnings  does  he 
save? 

5.  What  per  cent  of  $  36  is  $  24  ? 

6.  What  per  cent  of  the  cost  does  a  jeweler  make  by 
selling  a  watch  for  $  20  that  cost  him  $  14  ? 

7.  Of  what  sum  is  60  dollars  621%  ? 

8.  A  cow  cost  $45,  which  was   15%  of  the  cost  of  a 
horse.     What  did  the  horse  cost  ? 

9.  A  dealer  sold  coal  at  $4.80  a  ton,  which  was  20% 
more  than  it  cost  him.     What  did  he  pay  for  it  ? 

10.  A  merchant  sold  a  pair  of  shoes  for  $  1.50,  thereby 
losing  25%  of  the  cost.     What  was  the  cost  ? 

11.  I  bought  a  horse  for  $200,  and  sold  it  at  an  ad- 
vance of  20%.     What  did  I  get  for  it  ? 

12.  What  per  cent  is  gained  by  selling  goods  that  cost 
10  cents  a  yard  at  12J  cents  a  yard  ? 

13.  An  agent  gets  a  discount  of  20%  from  the  retail 
price  of  articles,  and  sells  them  at  the  retail  price.     What 
is  his  gain  per  cent  ? 

14.  By  selling  butter  at  6  cents  a  pound  more  than  cost, 
a  grocer  made  20%.     What  did  he  pay  for  it  ? 


REVIEW  EXERCISES.  297 

15.  I  sold  two  cows  for  $45  each.     On  one  I  gained 
25%,  and  on  the  other  I  lost  25%.     Did  I  gain  or  lose  by 
the  transaction,  and  how  much  ? 

16.  A  boy  bought  apples  at  the  rate  of  3  for  5  cents, 
which  he  sold  at  the  rate  of  4  for  10  cents.     What  per  cent 
did  he  gain  ? 

17.  A  boy  lost   80   cents,  which  was  just  20%  of  his 
money.     How  much  money  had  he  ? 

18.  B  gained  $18,  which  was  30%  of  what  he  then  had. 
How  much  had  he  at  first  ? 

19.  A  merchant  bought  125  barrels  of  flour,  and  after 
losing  20%  of  it,  he  sold  25%  of  the  remainder.     What  per 
cent  of  the  whole  had  he  left  ? 

20.  A  cistern  containing  60  bbl.  of  water  receives  by 
one  pipe  5%  of  its  contents  in  an  hour,  and  by  another 
loses  15%.    How  much  remains  in  it  at  the  end  of  an  hour  ? 

21.  When  a  man  sells  goods  at  a  price  from  which  he 
received  a  discount  of  40%,  what  is  his  gain  per  cent  ? 

22.  What  per  cent  does  a  merchant  gain  who  buys  flour 
at  $  4.50  a  barrel,  and  sells  it  at  $  6  a  barrel  ? 

23.  A  book  was  sold  for  90  cents,  which  was  at  a  gain  of 
20%.     What  would  have  been  the  gain  per  cent  if  it  had 
been  sold  for  $  1  ? 

24.  An  agent  sold  $870  worth  of  goods  at  a  commission 
of  3^%.     How  much  did  he  receive  ? 

25.  A  book  agent  received  $60  for  selling  $150  worth 
of  books.     What  was  his  rate  of  commission  ? 

26.  A  real  estate  agent  received  $  120  for  selling  a  house 
and  lot,  at  2%  commission.     For  how  much  was  the  prop- 
erty sold  ? 

27.  A  company  declares  a  dividend  of  121% .    How  much 
will  a  stockholder  owning  20  shares  receive  ? 


298  INTEREST. 

28.  An  agent  received  $  324  with  which  to  buy  peaches, 
after  deducting  his  commission  of  8%.     How  much  did  he 
expend  for  peaches  ? 

29.  A  house  worth  $6000  was  insured  for  f  of  its  value, 
at  1-^%.     What  was  the  premium  ? 

30.  What  is  the  interest  of  $300  for  3  years  4  months, 
at  6%  ? 

WRITTEN  EXERCISES. 

399.  1.  The  number  of  youth  of  school  age  in  a  certain 
city  is  16,767,  which  is  34^-%  of  the  number  of  inhabitants. 
What  is  the  population  of  the  city  ? 

2.  A  man  invested  $8160  in  land,  which  was  62-J-%  of 
all  his  money.     How  much  money  had  he  left  ? 

3.  A  farm  was  sold  for  $  6300,  which  was  12|-%  more 
than  it  cost.     What  was  the  cost  of  the  farm  ? 

4.  By  selling  wheat  at  a  gain  of  15%,  a  speculator  re- 
ceived $  20,125.     What  did  the  wheat  cost  him  ? 

5.  If  a  teacher  who  receives  a  yearly  salary  of  $  900  pays 
$  250  a  year  for  board,  and  $  100  for  other  expenses,  what 
per  cent  of  his  salary  does  he  save  ? 

6.  A  speculator  had  6000  barrels  of  flour  that  cost  him 
$4.50  a  barrel.     He  sold  30%  of  the  lot  at  an  advance  of 
10  fo  of  the  cost,  and  50  </0  of  the  remainder  at  an  advance 
of  12-j-$  of  the  cost.     He  then  closed  out  the  lot  at  $5  a 
barrel.     How  much  did  he  gain  on  the  flour  ? 

7.  A  man  left  $4500  to  his  wife,  which  was  62£%  of 
the  sum  bequeathed  to  his  children,  and  the  sum  bequeathed 
to  his  wife  and  children  was  75%  of  his  estate.     What  per 
cent  of  the  estate  did  the  wife  receive  ? 

8.  A  drover  bought  horses  at  $145  a  head,  paid  $11 
for  taking  each  of  them  to  market,  and  then  sold  them  at 
$  175.50  a  head.     What  was  the  gain  per  cent  ? 


REVIEW  EXERCISES.  299 

9.    A  man  paid  $  4860  for  a  farm,  and  sold  it  for  $  5346. 
What  was  the  gain  per  cent  ? 

10.  An  agent  in  Savannah  received  $12,180  with  which 
to  purchase  cotton,  after  deducting  his  commission,  of  1^%. 
How  much  did  he  expend  for  cotton,  and  what  was  his ' 
commission  ? 

11.  What  will  be  the  total  cost  of  800  yards  of  carpet- 
ing, at  $1.60  a  yard,  if  a  merchant  pays  1J%  commission 
for  purchasing  it  ? 

12.  A  company  with  a  capital  of  $  76,500  declares  a  divi- 
dend of  7%,  and  still  has  a  surplus  of  $2500.     What  were 
the  net  earnings  of  the  company  ? 

13.  The  owners  of  a  ship  paid  $306  for  an  insurance 
policy  of  $  13,600.     What  was  the  rate  of  premium  ? 

14.  A  grain  dealer  paid  $225  for  insuring  a  cargo  of 
wheat  at  l-J-%-     For  how  much  was  it  insured  ? 

15 .  For  what  sum  must  a  cargo  of  goods  valued  at  $12,360 
be  insured,  at  If  %,  to  cover  both  property  and  premium,  in 
case  of  loss  ? 

16.  If  the  assessed  valuation  of  a  town  is  $  2,360,000,  and 
the  town  has  640  polls,  paying  $  1.50  each,  what  must  be 
the  rate  of  taxation  in  order  to  raise  $  10,400  ? 

17.  A  manufacturer  imported  from    Spain   30  bales  of 
wool,  300  Ib.  each,  invoiced  at  32  cts.  per  pound,  and  25 
bales,  250  Ib.  each,  invoiced  at  30  cts.  per  pound.     What 
was  the  duty,  at  20%  ad  valorem  ? 

18.  A  speculator  bought  65  shares  W.  U.  Telegraph  stock 
at  90,  and  sold  them  at  94|.     How  much  did  he  gain,  bro- 
kerage in  each  case  being  -§•%  ? 

19.  A  man  borrowed  $160  Apr.  1,  1889,  and  paid  it  Dec. 
8,  1891,  with  interest  at  6%.     What  amount  did  he  pay  ? 

20.  My  taxes  were   $  315.25.     What  was   the   assessed 
valuation  of  my  property,  if  the  rate  of  taxation  was  .015  ? 


300  INTEREST. 

21.  A  note  of  $300,  dated  May  12,  1887,  is  due  in  4 
years,  with  interest  at  6%,  payable   annually.      If  both 
interest  and   principal   remain   unpaid,  what  will   be   the 
amount  due  on  the  note  May  12,  1891  ? 

22.  What  is  the  compound  interest  of  $  750  for  3  yr. 
8  mo.  12  da.,  at  6%  ? 

23.  What  principal  will  yield  $13.50  interest  in  9  mo. 
18  da.,  at  6%  ? 

24.  What  is  the  difference  between  the  present  worth 
and  proceeds  of  $  560  due  in  2  yr.  6  mo.,  at  6%  ?    (No  grace.) 

25.  A  grain  speculator  bought  6000  bushels  of  wheat,  at 
95  cents  per  bushel,  cash.     He  sold  it  the  same  day  at  an 
advance  of  3|-%,  receiving  in  payment  a  note  due  in  1  mo., 
without  interest,  which  he  had  discounted  at  a  bank  at  6%. 
What  was  his  gain  in  cash  ? 

26.  A  piano,  the  list  price  of  which  was  $  420,  was  sold 
at  a  discount  of  30%  and  10%.     If  the  freight  was  $6.50, 
and  the  dray  age  $2.75,  what  was  the  net  cost  of  the  piano  ? 

27.  What  was  the  list  price  of  an  article  whose  net  cost 
was  $4.50,  after  deducting  discounts  of  40%  and  10%  ? 

28.  What  are  the  net  proceeds  of  a  sale  of  300  bbl.  pork 
at  $  20  per  barrel,  less  the  following  charges  :  freight,  40^ 
per  barrel ;  insurance,  \°/0  '•>  commission,  2-^%  ? 

29.  After  getting  a  note,  without  interest,  discounted  at 
a  bank  for  3  mo.  at  6%,  I  had  $  354.42.    What  was  the  face 
of  the  note  ?     (Allow  days  of  grace.) 

30.  I  sold  -f  of  my  property  for  cash,  at  a  gain  of  33^%, 
and  the  rest  for  f  of  the  cost  of  the  whole,  receiving  in  pay- 
ment a  note  due  in  3  months,  without  interest,  which  I  got 
discounted  at  a  bank,  at  6%.     What  was  my  gain  per  cent, 
if  my  property  cost  $  24,000  ? 

31.  A  man  sold  144  shares  of  Mass.  5's  at  par,  and  in- 
vested the  proceeds  in  Mich.  7's  at  120.     What  was  the 
change  in  his  annual  income  ? 


EXCHANGE. 


400.  1.  When  A  owes  B  $  500,  and  B  owes  A  $500,  how 
may  the  accounts  be  settled  without  any  transfer  of  money 
taking  place  ? 

2.  When  A  in  Chicago  owes  B  in  New  York  $500,  and 
C  in  New  York  owes  A  $  1000,  how  can  A  pay  his  indebted- 
ness to  B  without  remitting  the  money  ? 

3.  What  will  be  the  indebtedness  of  A,  B,  and  C  to  each 
other  after  the  transaction  has  taken  place  ? 

4.  A  and  C  live  in  the  same  city,  and  B  in  a  distant  city. 
A  owes  B  $  2000,  and  B  owes  C  $  1000.     How  may  B  pay 
his  indebtedness  to  C  without  remitting  the  money  ? 

5.  What  will  be  their  indebtedness  to  each  other  after 
A  has  paid  B's  order,  or  draft  ? 

6.  What  will  a  draft  for  $500  cost,  payable  when  it  is 
presented,  or  at  sight,  if  %%  premium  is  charged  for  it  ? 

7.  How  much  should  be  deducted  from  the  price  of  the 
above  draft  if  it  is  not  to  be  paid  until  two  months,  money 
being  worth  6%  ? 

8.  What  will  be  the  cost  of  a  draft  for  $  50,  payable  at 
sight,  if  it  is  purchased  at  1%  discount? 

9.  What  will  be  the  cost  of  a  sight  draft  for  $  300,  pur- 
chased at  \<J0  premium  ? 

10.  If  A  in  Nashville  owes  B  in  New  Orleans  $  1000,  and 
C  in  New  Orleans  owes  D  in  Nashville  $  1500,  how  may  A 
pay  his  indebtness  without  remitting  the  money  ? 

301 


302  EXCHANGE. 

11.  If  the  premium  is  £%,  how  much  will  it  cost  me  to 
remit  a  draft  for  $  800  from  Cincinnati  to  Cleveland  ? 

12.  If  a  man  sells  a  draft  for  $  500,  at  a  premium  of  }%, 
how  much  does  he  receive  for  it  ? 

13.  A  wishes  to  send  to  his  agent  in  New  Orleans  a  draft 
for  $5000.     If  the  premium  on  exchange  is  -|%,  how  much 
will  the  draft  cost  him  ? 

14.  When  I  pay  $2025  for  a  sight  draft  on  New  York 
for  $  2000,  what  is  the  premium,  or  rate  of  exchange  ? 

15.  When  I  can  buy  a  sight  draft  on  Chicago  for  $2000, 
paying  for  it  $  1980,  what  is  the  rate  of  exchange  ? 

16.  If  Mr.  Burt  pays  $  4975  for  a  sight  draft  on  Cincinnati 
for  $  5000,  at  what  rate  is  exchange  ? 

401.  The  method  of  making  payments  in  distant  places 
without  transmitting  money  is  termed  Exchange. 

1.  Thus,  when  A  in  San  Francisco  owes  B  in  New  York  $500,  and 
C  in  New  York  owes  D   in   San  Francisco  $500,  C   may  go  to  B 
and  pay  him  $  500  for  an  order  upon  A  in  San  Francisco  for  $  500, 
and  then  send  it  to  D.     A  then  pays  D,  and  the  indebtedness  is  paid 
without  the  transmission  of  money. 

2.  Exchange  is  therefore  a  very  convenient  and  safe  way  of  can- 
celling debts. 

3.  The  business  is  carried  on  largely  by  banks,  which  charge  a  small 
sum  for  transacting  the  same. 

402.  The  written  order  of  one  party  to  another,  to  pay  a 
specified  sum  of  money  to  the  party  named  in  his  order,  is 
termed  a  Draft  or  Bill  of  Exchange. 

An  order  upon  a  bank,  by  a  person  who  has  money  deposited  in  it, 
is  called  a  Check. 

403.  The  person  who  makes  the  order  is  the  Drawer. 
The  person  to  whom  the  order  is  addressed  is  the  Drawee. 
The  person  to  whom  the  money  is  to  be  paid  is  the  Payee. 


DOMESTIC   EXCHANGE.  303 

404.  When  a  drawee  accepts  a  draft,  he  writes  the  word 
"  accepted"  upon  the  face  of  the  draft  with  the  date  of 
acceptance. 

405.  A  draft  made  payable  on  presentation  is  termed  a 
Sight  Draft.     A  draft   payable  at  a   specified  time   after 
presentation  or  after  sight  is  a  Time  Draft. 

The  laws  in  the  various  states  as  to  grace  on  drafts  are  not  strictly 
in  accord  with  those  concerning  grace  on  notes.  In  the  examples  given 
grace  is  allowed  on  time  drafts,  but  not  on  sight  drafts. 


FORM   OF  A  DRAFT. 

20, 


to  t/v&  o'bd&'b  o-^   Jb-cwid,  ff&nci&^o-n, 
i^ietu  — 

'    100 

to  tfa&  cuMs&wnt  o-i 


DOMESTIC  EXCHANGE. 

406.   Domestic  Exchange  treats  of  drafts  payable  in  the 
country  in  which  they  are  made. 

1.  When  the  bankers  of  Denver  have  not  sufficient  money  on 
deposit  in  New  York  to  meet  the  drafts  they  are  making  upon  New 
York,  they  must  send  money  to  meet  them.    This  naturally  raises  the 
price  of  drafts  in  Denver,  or  exchange  on  New  York  is  at  a  premium. 

2.  When  the  bankers  of  Denver  have  large  sums  of  money  de- 
posited in  the  banks  of  New  York  upon  which  they  receive  no  interest, 
they  are  often  anxious  to  sell  drafts  on  New  York  at  a  discount,  so 
that  they  may  get  money  to  use  at  home  without  the  expense  of  having 
it  forwarded  by  express. 


304  EXCHANGE. 

WRITTEN  EXERCISES. 

407.  1.  What  will  be  the  cost  of  a  sight  draft  upon  New 
York  for  $10,000,  at  \%  premium  ? 

EXPLANATION.  —  Since    exchange 
$  1  -f  $  .00^=$  l.OOJ.       on  New  York  is  at  i  %  premium,  every 

«1  nni  vin  noo    rcm  09*    dollar  of  the  draft  wil1  cost  $1-°°*' 
$  1.00  J  x  10,000  =  $  10,025.    and  a  draft  for  $  10)000  will  cost  10)000 

times  1.00£,  or  $10,025. 

2.  What  will  be  the  cost  in  Boston  of  a  draft  for  $5000 
on  St.  Paul,  payable  2  mo.  after  date,  the  rate  of  exchange 
being  at  \°/0  premium  ? 

$  1  +  $  ,00i=  $  1.005.  EXPLANATION.  —Since  the  ex- 

$  1.005-$  .0105=$  .9945. 


$  .9945  x  5000  =  $  4972.50.     would  cost  $  i-005  if  Paid  at  sight- 

But  since  the  draft  is  not  to  be 

paid  in  St.  Paul  for  2  mo.  and  3  da.,  the  banker  in  Boston,  who  has 
the  use  of  the  money  for  that  time,  inasmuch  as  he  is  not  obliged  to 
pay  the  draft  for  2  mo.  3  da.  ,  allows  the  bank  discount  on  the  face  of 
the  draft  for  that  time.  The  bank  discount  in  Minnesota  for  that  time 
at  the  legal  rate  is  $  .0105,  and  this  subtracted  from  $  1.005  gives  the 
cost  of  $  1  of  the  draft.  Since  the  cost  of  $1  of  the  draft  is  $  .9945, 
the  cost  of  $  5000  is  5000  times  that  sum,  or  $  4972.50. 

3.  What  will  be  the  cost  in  St.  Louis,  Mo.,  of  a  sight 
draft  on  Philadelphia  for  $  1200,  the  rate  of  exchange  being 
at  |%  premium? 

4.  What  will  be  the  cost  in  Denver  of  a  sight  draft  on 
Boston  for  $  1500,  exchange  being  at  |%  discount? 

5.  What  must  be  paid  in  New  York  for  a  sight  draft  on 
Cleveland  for  $  800,  when  exchange  is  at  |%  premium  ? 

6.  What  will  be  the  cost  in  Cincinnati  of  a  draft  for 
$  1600  on  Topeka,  payable  2  mo.  after  date,  exchange  being 
at  \%  premium,  and  interest  at  8%  ? 

7.  What  must  be  paid  for  a  draft  for  $  475,  payable  30 
days  after  date,  at  \°/o  premium,  and  interest  at  6%  ? 


DOMESTIC   EXCHANGE.  305 

8.  What  must  be  paid  for  a  draft  drawn  at  Philadelphia 
on  Indianapolis  for  $600,  payable  90  days  after  date,  at 
\o/0  discount,  interest  at  6%  ? 

9.  Find  the  cost  of  a  draft  for  $  900,  payable  in  60 
days,  when  exchange  is  at^-%  premium,  and  interest  at  7%. 

10.  What  will  be  the  cost  of  a  draft  on  Galveston  for 
$1200,  payable  in  60  days,  exchange  being  at  1^%  dis- 
count, and  interest  at  8%  ? 

11.  What  must  be  paid  for  a  draft  of  $550,  at  30  days, 
exchange  being  at  f  %  premium,  and  interest  at  4%  ? 

12.  Find  the  cost  of  a  draft  on  Des  Moines  for  $1750, 
payable  90  days  after  date,  exchange  being  at  1^%  discount, 
and  interest  at  7%. 

13.  How  large  a  sight  draft  on  New  Orleans  can  be  pur- 
chased for  $ 5000,  when  the  exchange  is  at  1|%  premium? 

$  1  +  $  .015  =  $1.015.  EXPLANATION. -Since  exchange  is 

«  *nnn  •  « 1  m  *  AQOK  n  at  1i%  Premiurn>  it  will  cost  $  1.015 
$50CO-^$1.015=492b.ll  to  buy  a  draft  for  ^  and  $5000 

will  buy  a  draft  for  as  many  dollars  as  $  1.015  is  contained  times  in 
$5000,  or  $4926.11. 

14.  How  large  a  draft  in  Buffalo,  N.Y.,  can  be  purchased 
for  $  3000,  payable  2  mo.  after  sight  in  Ealeigh,  KG.,  ex- 
change being  at  1  %  discount  ? 

<K  1  HO     <K  m  —  <K  QQ  EXPLANATION.  —  Since  exchange 

is  at  1  %  discount,  it  would  cost  $  .99 
$  .99— $  .0105  =  $  .9795.        to  buy  a  draft  of  $  1,  if  it  were  pay- 

$3000-^$. 9795=3062.78+    able  at  sisht-    But  since  tne  draft 

is  not  to  be  paid  until  2  mo.  3  da., 

the  banker  in  Buffalo,  who  has  the  use  of  the  money  for  that  time, 
allows  bank  discount  upon  the  face  of  the  draft  for  that  time,  or  $  .0105 
for  every  dollar.  Therefore,  since  it  costs  $  .9795  to  purchase  a  draft 
of  $1,  $3000  will  purchase  a  draft  for  as  many  dollars  as  $.9795  is 
contained  times  in  $  3000,  or  $  3062.78. 

15.  How  large  a  sight  draft  can  be  purchased  on  Cincin- 
nati for  $  2800,  when  the  rate  of  exchange  is  at  f  %  premium  ? 

16.  How  large  a  sight  draft  can  be  bought  on  Boston, 
Mass.,  for  $  1260,  when  the  exchange  is  at  ^\%  premium? 

STAND.    AR. 20 


306  EXCHANGE. 

17.  How  large  a  sight  draft  on  New  York  can  be  pur- 
chased in  St.  Joseph,  Mo.,  for  $  1800,  when  the  exchange  is 
at  |%  discount  ? 

18.  How  large  a  draft,  payable  30  days  after  sight,  can 
be  bought  for  $  2000,  exchange  being  at  1  %  premium,  and 
money  being  worth  6%  ? 

19.  Find  the  face  of  a  draft  on   Detroit,    at    60    days 
sight,  bought  for  $650,  exchange  being  at  1|%  premium, 
and  money  being  worth  6%. 

20.  What  is  the  face  of  a  draft  on  New  Orleans,  at  90 
days  sight,  which  may  be  bought  for  $  1000,  exchange  being 
at  |-%  discount,  and  money  being  worth  7%  ? 

FOREIGN  EXCHANGE. 

'   408.   Foreign   Exchange  treats  of   drafts   drawn   in   one 
country  and  payable  in  another. 

1.  Drafts  drawn  in  one  State  and  payable  in  another  are  some- 
times considered  as  foreign  drafts  or  bills  of  exchange. 

2.  Foreign  bills  of  exchange  are  drawn  upon  Antwerp,  Amsterdam, 
Hamburg,  Bremen,  Berlin,  and  other  commercial  centers,  but  drafts 
upon  London  and  Paris  are  much  more  common  since  they  are  paid 
in  any  part  of  Europe. 

409.  Three  drafts  or  bills  of  the  same  date  and  tenor, 
named  respectively  the  first,  second,  and  third  of  exchange 
are  sent  by  different  mails,  so  that  if  one  is  lost  the  other 
may  be  presented.     Such  a  set  is  called  a  Set  of  Exchange. 

When  one  bill  of  the  set  is  paid  the  others  are  void. 

410.  The  value  of  a  pound  sterling  or  sovereign  in  Ameri- 
can gold  is  $  4.8665. 

The  value  of  a  franc  is  about  $  .193,  or  5.18  francs  per 
dollar. 

The  values  of  the  pound  sterling  and  the  franc  given  above,  are  the 
values  when  exchange  is  at  par,  but  they  are  continually  fluctuating 
on  account  of  the  demand  for  bills  of  exchange,  the  rate  being  above 
or  below  par  according  as  the  demand  is  large  or  small. 


FOREIGN  EXCHANGE.  307 

WRITTEN   EXERCISES. 

411.  1 .  What  is  the  cost  in  New  York  of  a  sight  draft  on 
London  for  £312  15s.  5<i,  when  exchange  is  $4.87  for  a 
pound  sterling  ? 

SOLUTION. 

£312  15s.  5d.  =  £312.7708,  value  in  pounds  and  decimals  of  a  pound. 
$4.87  x  312.7708  =  $1523.193  +,  the  cost  of  the  draft. 

2.  How  large  a  bill  of  exchange  at  sight  on  London  can 
be  bought  in  New  York  for  $2984.38,  exchange  being  at 
$  4.86  for  a  pound  sterling  ? 

SOLUTION. 

$2984.38  ~  $4.86  =  614.0699.     .-.  $2984.38  =  £614.0699. 
£614.0699  =  £614  Is.  4 1 d,  the  face  of  the  draft. 

3.  What  must  be  paid  in  New  York  for  a  bill  of  ex- 
change  at  sight  on  London  for  £425  8s.,  when  sterling 
exchange  is  quoted  at  $4.87-^-? 

4.  What  will  a  stemng  bill  at  sight  for  £317  9s.  cost 
in  Philadelphia  when  exchange  is  quoted  at  $  4.90J  ? 

5.  How  large  a  draft  at  sight  on  London  can  be  bought 
in  Chicago  for  $  1950,  when  exchange  is  $  4.86J  ? 

6.  How  large  a  bill  of  exchange  at  sight  on  London  can 
be  bought  in  New  York  for  $2875.80,  when  exchange  is 
quoted  at  $4.871? 

7.  How  large  a  bill  of  exchange  at  sight  on  London  can  be 
bought  in  New  York  for  $  4000,  when  exchange  is  $  4.865  ? 

8.  How  much  must  be  paid  for  a  bill  of  exchange  on 
Paris,  at  sight,  for  5000  francs,  exchange  being  5.16  francs 
to  the  dollar  ? 

9.  What  must  be  paid  for  a  sight  bill  of  exchange  on 
Paris  for  7865  francs,  exchange  being  5.18  francs  to  the 
dollar  ? 

10.  Find  the  cost  of  a  bill  of  exchange  at  sight  on  Bre- 
men for  5344  marks,  exchange  being  at  $  .95  (per  4  marks). 


PARTNERSHIP. 


412.  1.  If  two  men,  who  have  equal  sums  invested  in 
the  same  business,  gain  $  100,  what  is  each  man's  share  of 
the  gain? 

2.  If  one  man  furnishes  •§•  of  the  capital,  and  another  ^ 
of  it,  and  the  gain  is  $  1200,  what  should  be  the  gain  of  each  ? 

3.  Mr.  A  furnishes  $  3000  of  the  capital,  and  Mr.  B.  fur- 
nishes the  rest,  which  is  $  5000.     What  part  of  the  profits 
should  each  receive  ? 

4.  Four  partners  furnish  money  in  the   proportion   of 
$  2000,  $  3000,  $  4000,  and  $  5000  respectively.     What  part 
of  the  gain  should  each  one  receive  ? 

5.  Three  men  engage  in  business  and  furnish  the  follow- 
ing sums  respectively:   A,  $ 5000;    B,  $4000;   C,  $3000. 
How  much  of  the  gain  should  each  receive,  if  $  1200  was 
gained  during  the  year  ? 

6.  The  profits  of  a  company  were  $800  for  a  certain 
time.     What  share  of  the  profits  did  each  partner  receive, 
if  the  capital  contributed  by  them  was  $900,  $700,  and 
$800  respectively? 

7.  A  and  B  formed  a  partnership  after  A  had  been  doing 
business  alone  for  6  months.    A  had  $  5000  invested  during 
the  year,  and  B  had  $10,000  invested  for  6  months.     The 
gain  was  $  5000.     What  was  each  one's  share  of  the  gain  ? 

8.  The  cost  of  a  pasture  was  $27.     A  had  in  it  5  cows 
for  3  weeks,  and  B  3  cows  for  4  weeks.     What  should  each 
one  pay  ? 


WRITTEN  EXERCISES.  309 

413.  An  association  of  two  or  more  persons  for  the  pur- 
pose of  conducting  business  is  a  Partnership. 

414.  The  persons   associated  in  business   are  Partners. 
They  are  termed  collectively  a  company,  a  firm,  or  a  house. 

415.  PRINCIPLE.  —  The  gains   and  losses  of  a  firm  are 
shared  in  proportion  to  the  amount  of  capital  each  partner 
invests,  and  the  length  of  time  it  is  used  in  the  business. 

WRITTEN    EXERCISES. 

416.  1.  A,  B,  and  C  engaged  in  business,  A  furnishing 
$9000  of  the  capital,  B  $5000,  and  C  $6000.      If  they 
gained  $  6000,  what  was  each  partner's  share  of  the  gain  ? 

SOLUTION. 
$  9000  +  $  5000  +  $  6000  =  $  20,000,  the  entire  capital. 

or  _9^  =  A's  share  of  the  capital, 
of  $  6000  =  $  2700,  A's  share  of  the  gain. 

or  i  =  B's  share  of  the  capital. 
.-.    \  of  $  6000  =  $  1500,  B's  share  of  the  gain. 

•fffifo  or  -j-%  =  C's  share  of  the  capital. 
...  ^  of  $6000  =  $  1800,  C's  share  of  the  gain. 

2.  A/B,  and  C  formed  a  partnership  in  business,  A  fur- 
nishing $  8000  of  the  capital,  B  $  4500,  and  C  $  3500.     They 
gained  $3200   the  first  year.     What  was   each   partner's 
share  of  the  gain  ? 

3.  A,  B,  and   C   formed   a   partnership,  A   putting  in 
$4500,  B  $5400,  and  C  $4200.     On  closing  the  business 
they  found  they  had  lost  $2400.     What  was  the  loss  of 
each  ? 

4.  Three  men  engaged  in  business.     A  furnished  $  6000 
of  the  capital,  B  $9600,  and  C  $6400.     They  made  a  net 


310  PARTNERSHIP. 

gain  of  $  4800,  and  then  sold  out  for  $  30,000.     What  was 
each  partner's  share  of  the  gain  ? 

5 .  A,  B,  C,  and  D  formed  a  partnership.    A  put  in  $  5625, 
B  $  5250,  C  $  7125,  and  D  $  6000.     What  was  each  part- 
ner's share  of  a  profit  amounting  to  $  6960? 

6.  A,  B,  and  C  formed  a  partnership,  A  contributing 
$5500,  B  $6500,  and  C  $4500.     When  the  business  was 
closed  up   C   received   $1500   for   his  share  of  the  gain. 
How  much  should  each  of  the  others  receive  ? 

7.  A  and  B  were  engaged  in  business  two  years,  making 
an  annual  profit  of  $  8190.     During  the  first  year  A  owned 
-J  of  the  stock,  and  during  the  second  year  B  owned  f  of  it. 
What  was  each  partner's  share  of  the  total  profits  ? 

8.  E,  F,  G,  and  H  formed  a  partnership  with  a  capital 
of  $30,000.     E  furnished  $6000,  F  $7000,  G  $8000,  and 
H  the  remainder.     They  gained   18%   of  the  joint  stock. 
What  was  each  partner's  share  of  the  profit  ? 

9.  Three  partners  had  a  gain  of  $  6250  to  divide  accord- 
ing to  each  member's  investment.     A  invested  $  10,000,  B 
invested  $  15,000,  and  C  invested  $25,000.     What  was  the 
net  gain  of  each  ? 

10.  A,  B,  and  C  engage  in  business  together.  A  puts  in 
$  20,000,  and  after  3  mo.  he  takes  in  B  as  a  partner  with 
$  20,000.  At  the  end  of  3  mo.  more  they  take ,  in  C  as 
a  partner,  with  a  capital  of  $10,000.  If  they  gained 
during  the  year  $  7800,  what  was  each  partner's  share  of 

the  gain  ? 

SOLUTION. 

$20,000  employed  for  12  mo.  =  $240,000  for  1  mo.,  A's  capital. 
$20,000  employed  for  9  mo.  =  $  180,000  for  1  rno.,  B's  capital. 
$  10,000  employed  for  6  mo.  =  $  60,000  for  1  mo. ,  C's  capital. 

The  entire  capital  for  1  mo.  =  $480,000 

.-.  A's  share  of  the  gain  is  ff${fj$  or  J  of  $  7800,  which  is  $3900. 
B's  share  of  the  gain  is  if$$$  or  f  of  $  7800>  which  is  $2925. 
C's  share  of  the  gain  is  $$ft&  or  £  of  $  7800,  which  is  $975. 


WRITTEN   EXERCISES.  311 

11.  A,  B,  and  C  entered  into  partnership.      A  put  in 
$2800  for  10  months,  B  $3200  for  1  year,  and  C  $4000  for 
8  months.    They  gained  $  2952.    What  was  each  one's  share 
of  the  gain  ? 

12.  A  and  B  entered  into  partnership  for  one  year.     A 
had  $  1200  in  the  business  during  the  first  four  months,  and 
$  800  more  during  the  remainder  of  the  year.     B  had  $  1000 
during  the  first  six  months,  and  then  took  out  $  200.     At 
the  end  of  the  year  they  found  they  had  lost  $  1580.    What 
was  each  partner's  loss  ? 

13.  A,  B,  and  C  hired  a  pasture  for  $128.     A  pastured 
6  horses  for  8  weeks,  B  12  oxen  for  10  weeks,  and  C  40  cows 
for  12  weeks.     If  2  horses  eat  as  much  as  3  oxen,  and  3 
oxen  as  much  as  5  cows,  how  much  did  each  man  pay  ? 

14.  A,  B,  and  C  engaged  in  business.     A's  capital  was  in 
trade  4  months,  B's  5  months,  and  C's  12  months.     A's  gain 
was  $  800,  B's  $  1000,  and  C's  $  1200,  and  the  whole  capital 
was  $  25,675.     How  much  capital  did  each  furnish  ? 

15.  D  and  E  rented  a  pasture  for  $480.     D  put  in  400 
sheep,  and  E  320.     At  the  end  of  4  months  they  disposed 
of  half  their  stock,  and  allowed  F  to  put  in  240  sheep. 
What  rent  should  each  pay  at  the  end  of  the  8  months  ? 

16.  A,  B,  and  C  formed  a  partnership.     A  put  in  $3000 
for  5  months,  and  then  increased  it  $  1500  for  4  months 
more.     B  put  in  $  9000  for  4  months,  and  then  withdrawing 
half  his  capital,  continued  the  remainder  3  months  longer. 
C  put  in  $  5500  for  7  months.     They  gained  $  3630.     What 
was  each  partner's  share  of  the  gain  ? 

c  17.  A  and  B  entered  into  partnership  Jan.  1,  each  fur- 
nishing $3000  capital.  Apr.  1,  A  added  $500,  and  Sept.  1, 
he  added  $500  more.  June  1,  B  added  $1000.  What 
share  of  the  profits  should  each  receive  at  the  end  of  the 
year,  if  they  gained  $  2500  ? 


RATIO. 


417.  1.    How  does  $  3  compare  with  $  6  ?   $  5  with  $  15  ? 
$8  with  $24? 

2.  How  does  2  compare  with  12  ?   3  with  18  ?   5  with  25  ? 

3.  What  relation  is  2  to  10  ?   5  to  25  ?   6  to  30  ? 

4.  What  relation  is  4  to  12  ?   10  to  40  ?   6  to  36  ? 

5.  How  does  12  compare  with  2  ?   18  with  3  ?   25  with  5  ? 

6.  How  does  18  compare  with  6  ?  15  with  5  ?  40  with  10  ? 

7.  What  is  the  relation  of  4  to  8  ?     Between  4  and  8  ? 

8.  What  is  the  relation  of  12  to  4  ?     Between  12  and  4  ? 

418.  The  relation  of  one  number  to  another  of  the  same 
kind  is  Ratio. 

1.  This  relation  may  be  expressed  in  two  ways :  thus,  when  it  is 
asked,  "  what  is  the  relation  of  4  to  8  ?  "  the  answer  may  be  4  is  £  of 
8,  called  the  geometrical  ratio,  or  4  is  4  less  than  8,  called  the  arith- 
metical ratio. 

2.  When  the  relation  of  one  number  to  another  is  sought,  the  first 
number  is  the  dividend  and  the  second  the  divisor. 

3.  When  the  relation  between  two  numbers  is  sought,  either  may 
be  regarded  as  dividend  or  divisor. 

419.  The  numbers  compared  are  the  Terms  of  the  Ratio. 

I 

420.  The  first  term  is  the  Antecedent   and  the  second  the 
Consequent. 

Thus   in  the  ratio  of  5  to  10,   5  is  the  antecedent  and  10  the 
consequent. 

312 


EXERCISES.  313 

421.  The  colon  (:)  is  the  Sign  of  ratio. 

Thus,  the  ratio  of  6  to  15  is  expressed  6 :  15. 

Since  the  ratio  of  one  number  to  another  is  expressed  by  the  quo- 
tient arising  from  dividing  the  antecedent  by  the  consequent,  the  colon 
may  be  regarded  as  the  sign  of  division  without  the  dividing  line. 

422.  The  antecedent  and   consequent   together   form  a 
Couplet. 

EXERCISES. 

423.  What  is  the  ratio  of : 

1.  3  to    6?   15  to  30?  9  to  18?  54  to    6? 

2.  5  to  10?   20  to  10?  7  to  21?  28  to    4? 

3.  8  to  16?    16  to    4?  6  to  42?  33  to    3? 

4.  12  to  36?   36  to    6?  8  to  24?  56  to    7? 

5.  18  to  36  ?   40  to    8  ?  18  to  54  ?  72  to  12  ? 

6.  15  to  45  ?   35  to    7  ?  14  to  56  ?  85  to  17  ? 

What  is  the  ratio  of : 

7.  |  to  |? 

SUGGESTION.  —  The  ratio  of  f  to  f  is  the  same  as  the  ratio  of  2  to  4. 

s.  T\toA?  A  to  ,v?  Ato||?  ji  to     ? 
9.  A*°tt?  tttoA?  *to^?  ^to 

10.  /rtotf?  tftoA-?  A  to  |f?  |f  to 

What  is  the  ratio  of : 

11.  -|  to  |? 

SUGGESTION.  —  |  =  T\,  and  f  =  T9j  ;  therefore,  the  ratio  of  f  to  f  is 
the  ratio  of  T8j  to  T9^,  or  the  ratio  of  8  to  9  or  f . 

12.  |  to   |?      f  to   |?      \  to   f  ?      |  to  |? 

13.  \  to  |?      f  to  A?     *  to   f  ?      |  to  f? 

14.  A  to  A?   A  *>  H?    At 


PEOPOETION. 


424.     1.    Name  two  numbers  having  the  relation  to  each 
other  that  5  has  to  10. 

2.  Name  two  numbers  having  the  relation  that  4  has  to 
12.     3  to  15. 

3.  Name  two  numbers  having  the  relation  that  6  has  to 
30.     10  to  40. 

Name  two  numbers  having  the  relation  of : 

4.  5  to  20.     4  to  24.     7  to  21.       f  to  £. 

5.  6  to  24.     6  to  36.     5  to  40.       f  to  f . 

6.  7  to  35.     8  to  32.     9  to  27.      f  to  f 

7.  8  to  48.     7  to  56.     6  to  54.       f  to  f. 

8.  9  to  72.     8  to  72.     9  to  108.     £  to  f 

9.  What  number  has  the  relation  to  8  that  5  has  to 
20? 

What  number  has  the  relation 

10.  To  24  that  6  has  to  12  ?     To  15  that  £  has  to  £  ? 

11.  To  30  that  7  has  to  21  ?     To  25  that  -f  has  to  f  ? 

12.  To  48  that  9  has  to  36  ?     To  32  that  f  has  to  f  ? 

13.  To  72  that  5  has  to  60  ?     To  48  that  f  has  to  |  ? 

14.  To  88  that  8  has  to  64  ?     To  70  that  f  has  to  ^  ? 

15.  If  the  earnings  of  6  men  are  $30  in  a  given  time, 
how  will  the  earnings  of  10  men  compare  with  the  earnings 
of  6  men  in  the  same  time  ? 

314 


DEFINITIONS.  315 

• 

16.  Since  a  ratio  is  the  quotient  of  the  antecedent  divided 
by  the  consequent,  or   a  fraction,   what  changes   may  be 
made  upon  it  without  changing  its  value  ? 

17.  Write  two  equal  ratios.     Multiply  the  first  term  by 
the  last  term,  and  compare  their  product  with  the  product 
of  the  intermediate  terms. 

425.  An  equality  of  ratios  is  a  Proportion. 
Thus,  8  :  16  as  4  :  8  is  a  proportion. 

426.  A  double  colon  (: :)  is  the  Sign  of  proportion. 

It  is  written  between  the  ratios. 

It  has  sometimes  been  regarded  as  the  extremities  of  the  sign  of 
equality  (  =  ). 

The  sign  of  equality  is  often  used  in  proportion  instead  of  the 
double  colon. 

427.  A  proportion  must  have  four  terms,  viz.  two  ante- 
cedents and  two  consequents.     When  any  three  are  given, 
the  other  may  be  found. 

428.  The  first  and  third  terms  of  a  proportion  are  the 
Antecedents   of  the   proportion,  and  the  second  and  fourth 
terms  are  the  Consequents. 

Thus  in  the  proportions  5  :  8  :  :  10  : 16,  5  and  10  are  the  antecedents, 
and  8  and  16  the  consequents. 

429.  The  first  and  last  terms  are  the  Extremes,  and  the 
second  and  third  terms  are  the  Means  of  a  proportion. 

Thus  in  the  proportion  10  :  12  :  :  5  :  6,  10  and  6  are  the  extremes, 
and  12  and  5  are  the  means. 

430.  PRINCIPLES.  —  1.    The  product  of  the   extremes  is 
equal  to  the  product  of  the  means. 

2.  The  product  of  the  extremes  divided  by  either  mean  gives 
the  other  mean. 

3.  The  product  of  the  means  divided  by  either  extreme  gives 
the  other  extreme. 


316 


PROPORTION. 


EXERCISES. 
431.   Find  the  term  that  is  wanting  in  the  following : 


1.    36:18:      12?? 

11      25- 

22  5  •  •    ?    •  5  4 

2.    27:54: 

?:8. 

12.        ?: 

5:    27    :12.5. 

3.    24:72: 

21:? 

13.        f: 

?:    12    :18. 

4.     ?  :16: 

18:9. 

14.        ?: 

10    :3J    :6f. 

5.      9:    ?: 

6:24. 

15.        |: 

f:     i   :? 

6.    20:    5: 

?:4. 

16.        f: 

£:      ?    :f. 

7.    30:20: 

18:? 

17.        f  : 

f:      *    :? 

8.    45  :  60  : 

9  -24 

18.        ^: 

9  .        7        .4 

9.    48:24: 

8:? 

19.    £?: 

£24::  30    :  6. 

10.    50:75: 

100:? 

20.    $4: 

$10::6bu.:? 

21.    $5:  $40::  9 

Ib  :  ? 

22.    $.16:  $.32: 

?  •  4  at. 

23.  $6:$15::?:75yd. 

24.  10  men:  14  men::  $20:? 

25.  ?:351b.  ::$4:$7. 

26.  15pwt.  :?::21:10. 

27.  3.5  A.  :  10.5  A. :  :  ?  :  18. 

28.  12  men  :  42  men  : :  16  days  :  ? 

29.  18  mi.  :  4  mi.  :  :  20  :  ? 

30.  16  horses  :  28  horses  : :  -f  :  ? 

31.  21  da.  :35da.  ::?:& 

32.  I :  | :  :  10  bu.  :  ? 

33.  f  :  5  :  :  ?  :  40. 

34.  4:|::?:10. 


SIMPLE   PROPORTION.  317 

SIMPLE  PROPORTION. 

432.  A  ratio  between  any  two  numbers  is  a  Simple  Ratio. 

433.  An  equality  of  two  simple  ratios  is  a  Simple  Pro- 
portion. 

Thus,  8 : 12  : :  16  : 24  is  a  simple  proportion. 

WRITTEN  EXERCISES. 

434.  1.   If  9yd.  of  silk  cost  $27,  what  will  18  yd.  cost? 

EXPLANATION.  —  It  is  evident  that  the  cost  of  18  yd.  is  greater  than 

the  cost  of  9  yd.,  conse- 
/i\  yn'  IQ  <?rr  quently  the  answer  sought 

(1)  y:18::Jf:    C  is  greater  than  $27. 

yd.    yd.       $       $  Arranging  $27,  and  the 

(2)  18  :  9  : :  ?  :  27.  answer  sought  as  a  couplet 

-109/7  °f  the  proportion,  the  other 

(1)  By  Art.  430,  ?  =       X       =  54.     couplet  of  the  proportion 

9  must  be  expressed  to  cor- 

respond with  the  couplet 

/o\   -D      \   *    Aon  o       18  X  27       KA       first  arranged.      That  is, 

(2)  By  Art.  430,  ?  =  &—  =  54.     in  proport£n  (1)  the  con: 

sequent  of  the  second  coup- 
let is  greater  than  the  antecedent,  therefore  the  consequent  of  the  first 
couplet  must  be  greater  than  the  antecedent,  and  the  couplet  is  written 
9  yd.  :  18  yd. 

Solving  according  to  Art.  430,  the  value  of  18  yd.  is  $>  54. 

In  proportion  (2) ,  the  proportion  may  be  interpreted  thus : 

Greater  :  less  :  :  Greater  :  less. 

2.    If  6  men  can  dig  a  ditch  in  48  da.,  in  what  time  can 
8  men  dig  it  ? 

SUGGESTION.  —  8  men  can  do  the  work  in  less  than  48  da.,  conse- 
quently the  answer  is  less  than  48  da.,  and  the  proportions  may  be 
expressed  as  follows : 

(1)  8  men  :  6  men  :  :  48  da.  :  ? 
That  is,  Greater  :  less  :  :  Greater  :  less. 

(2)  6  men  :  8  men  :  :   ?  :  48  da. 
That  is,  Less  :  greater  :  :  Less  :  greater. 


818  PROPORTION. 

KULE.  —  Select  the  number  which  is  the  same  kind  as  the 
answer,  and  from  the  conditions  of  the  problem  discover 
ivhether  the  answer  is  to  be  greater  or  less  than  that  number. 

Arrange  these  two  terms  as  a  couplet,  and  then  arrange  the 
terms  of  the  other  couplet  to  conform  to  the  conditions  of  the 
first  couplet. 

Divide  the  product  of  the  extremes  or  means  by  the  single 
extreme  or  mean.  The  result  will  be  the  term  sought. 

Use  cancellation  whenever  it  is  possible  to  do  so. 

3.  If  15  tons  of  hay  cost  $120,  how  much  must  be  paid 
for  25  tons  ? 

4.  If  the  interest  upon  a  sum  of  money  for  9  months  is 
$  318.69,  what  will  be  the  interest  for  11£  months  ? 

5.  A  can  do  a  certain  piece  of  work  in  18  days,  working 
10  hours  per  day.     In  how  many  days  can  he  do  the  same 
work,  by  laboring  14  hours  per  day  ? 

6.  If  sound  travels  6160  ft.  in  5£  seconds,  how  far  does 
it  travel  in  a  minute  ? 

7.  How  high  is  a  church  spire  whose  shadow  is  162  ft. 
long,  when  a  flag-staff  60  ft.  high  casts  a  shadow  72  ft.  long  ? 

8.  B  did  a  piece  of  work  in  18  days,  thereby  earning 
$  2.80  a  day.     What  would  he  have  earned  per  day,  had  he 
done  the  work  in  16  days  ? 

9.  If  $  600  yields  $  140  interest  in  a  certain  time,  what 
interest  will  $  750  yield  in  the  same  time  ? 

10.  A  farmer  sowed  6  bu.  of  grain  on  4f  acres.     How 
many  bushels,  at  this  rate,  would  he  need  for  a  field  con- 
taining 13^-  acres  ? 

11.  If  a  garrison  of  150  men  consumes  26  barrels  of  flour 
in  9  weeks,  how  many  barrels  will  it  consume  in  22  J  weeks  ? 

12.  If  165  bushels  of  potatoes  are  raised  on  1-^-  acres, 
how  many  bushels  can  be  raised  on  3J  acres  ? 


SIMPLE  PROPORTION.  319 

13.  If  2f  barrels  of  beef  cost  $20.75,  how  much  will  7£ 
barrels  cost  ? 

14.  If  it  requires  42  yards  of  carpet  which  is  f  of  a  yard 
wide  to  cover  a  floor,  how  many  yards  of  carpet  one  yard 
wide  will  be  needed  to  cover  the  same  floor  ? 

15.  Thirty  men  can  dig  a  ditch  in  20  days.     After  they 
have  been  digging  12  days,  how  many  more  men  must  be 
employed  to  finish  it  in  6  days  more  ? 

16.  At  the  time  when  a  man  5  ft.  8  in.  in  height  casts  a 
shadow  4  ft.  6  in.  long,  what  is  the  height  of  a  tree  that 
casts  a  shadow  46  ft.  6  in.  long  ? 

17.  Two  cog-wheels,  one  having  26  cogs,  and  the  other 
20  cogs,  run  together.     In  how  many  revolutions  of  the 
larger  wheel  will  the  smaller  gain  12  revolutions  ? 

18.  If  15  men  can  do  a  piece  of  work  in  36  days,  in  how 
many  days  can  they  perform  the  same  work  with  the  assist- 
ance of  9  men  more  ? 

19.  A  piece  of  work  can  be  done  in  40  days  by  25  men. 
After  18  days,  13  men  quit  work.     In  how  many  days  can 
the  rest  finish  the  work  ? 

20.  If  a  garrison  of  200  men  has  provisions  for  8  months, 
how  many  men  must  leave  at  the  end  of  5  months  that  the 
provisions  remaining  may  last  the  rest  8  months  longer  ? 

21.  If  it   requires  15  compositors  15  days  to  set  up  a 
book  of  675  pages,  how  many  days  will  they  need  to  set  up 
a  book  of  900  pages  ? 

22.  If  a  railway  train  runs  444  miles  in  8  hr.  40  min.,  in 
what  time  can  it  run  1060  miles,  at  the  same  rate  of  speed  ? 

23.  A  farmer  raised  405  bushels  of  beets  upon  1  A.  20 
sq.  rd.  of  land.     How  many  bushels  can  he  raise  upon  a 
field  containing  7  A.  85  sq.  rd.  ? 

24.  A  train  which  runs  35 \  miles  per  hour  leaves  Chicago 
at  8:25  A.M.     How  far  will  it  have  traveled  at  2:30  P.M. 


320  PROPORTION. 


COMPOUND  PROPORTION. 

435.  The  product  of  two   or   more  simple   ratios  is   a 
Compound  Ratio. 

436.  A  proportion  in  which  either   or  both  ratios   are 
compound  is  a  Compound  Proportion. 

WRITTEN    EXERCISES. 

437.  1.    If  6  men  can  mow  24  acres  of  grass  in  2  days  by 
working  10  hours  per  day,  how  many  days  will  it  take  7  men 
to  mow  56  acres  of  grass  by  working  12  hours  per  day  ? 

-EXPLANATION.  —  A  compound  proportion  is  one  involving  several 

conditions.  The  first  con- 
dition in  this  problem  is : 
If  6  men  can  mow  the 
grass  in  2  days,  how  long 
will  it  take  7  men  to  do 
the  work?  The  solution  of 
this  is  expressed  by  pro- 
portion (1).  The  second 
condition  is  :  If  the  men 
can  mow  24  acres  of  grass 

in  x  days'  how  long  a  time 
z  =  _  ^     will  it  take  them  to  mow 

Extremes  7  X  24  X  12  56  acres  ?  This  is  solved 

by  simple  proportion  in 

proportion  (2),  giving  y  days.  The  third  condition  is  :  If  the  work  can 
be  done  by  the  men  in  y  days  by  working  10  hours  per  day,  how  many 
days  will  be  required  to  do  the  work  if  they  work  12  hours  per  day  ? 
This  solved  by  simple  proportion,  by  proportion  (3),  gives  z  days. 

Since  every  proportion  is  an  equality  of  ratios,  the  products  of  these 
proportions  term  by  term  will  be  an  equality  of  ratios,  or  a  proportion. 
Since  x  and  y  appear  in  both  antecedent  and  consequent,  they  may  be 
omitted  from  the  product,  since  antecedent  is  the  same  as  numerator, 
and  consequent  the  same  as  denominator,  and  the  simple  proportions 
will  assume  the  form  of  (4).  Solving,  the  answer  is  found  to  be  3^  da. 

The  problem  may  be  stated  at  the  outset  as  in  proportion  (4)  by 
writing  for  the  third  term  the  oae  that  is  of  the  same  kind  as  the 
answer,  and  then  arranging  couplets  by  considering  their  relation  to 
the  answer  sought. 


COMPOUND   PROPORTION.  321 

RULE.  —  Use  for  the  third  term  the  number  which  is  of  the 
same  kind  as  the  answer  required. 

Arrange  the  other  couplets  according  to  the  relation  of  the 
third  term  to  the  answer  sought. 

The  product  of  the  means  divided  by  the  product  of  the 
extremes  will  be  the  answer. 

Problems  in  compound  proportion  are  readily  solved  by  what  is 
termed  the  cause  and  effect  method. 

Example  1,  stated  by  cause  and  effect,  is  as  follows  : 

1st  cause.  2d  cause.  1st  effect.          2d  effect. 

6  men    ^        7  men    ~\          c  r 

2  days    [•  :     ?  days    [•  :  :  •!  24  acres  :  •!  56  acres. 
10  hours  J      12  hours  J          I  I 

2.  If  11  men  build  45  rods  of  wall  in  6  days  of  10  hours 
each,  how  many  men  will  be  required  to  build  81  rods  of 
wall  in  12  days  of  11  hours  each  ? 

3.  Three   workmen  dig  a  ditch  20  rd.  long  and   3   ft. 
wide  in  10  days.     How  long  will  it  take  5  workmen  to  dig 
a  ditch  45  rd.  long  and  4  ft.  wide  ? 

4.  If  18  men  can  perform  a  piece  of  work  in  12  day?, 
how  many  men  could  perform  another  piece  of  work  4  times 
as  great  in  -|-  of  the  time  ? 

5.  If  the  freight  charges  on  125  cattle,  averaging  900 
pounds,  is  $ 200  for  150  miles,  what  should  be  the  charges 
on  275  cattle,  averaging  1200  pounds,  for  225  miles  ? 

6.  If  a  block  of  marble  7  ft.  long,  3  ft.  wide,  and  2  ft. 
thick  weighs  6930  lb.,  what  will  be  the  weight  of  a  block  of 
the  same  kind  10  ft.  long,  4  ft.  wide,  and  3  ft.  thick  ? 

7.  If  the  capacity  of  a  bin  24  ft.  long,  4£  ft.  wide,  and 
4f  ft.  deep  is  405  bushels,  what  is  the  capacity  of  a  bin 
16  ft.  long,  5  ft.  wide,  and  4J  ft.  deep  ? 

8.  If  it  costs  $  180  to  build  a  wall  60  ft.  long,  14  ft.  high, 
and  1  ft.  6  in.  thick,  what  will  it  cost  to  build  a  wall  200 
ft.  long,  18  ft.  high,  and  1  ft.  4  in.  thick  ? 

STAND.  AR.  —  21 


822  PROPORTION. 

,9.  If  15  men,  working  12  hr.  a  day,  can  hoe  60  acres  in 
20  days,  how  long  will  it  take  35  boys,  working  10  hr.  a  day, 
to  hoe  90  acres,  the  work  of  5  men  being  equal  to  that  of  7 
boys? 

10.  If  16  men  can  excavate  a  cellar  40  ft.  long,  36  ft. 
wide,  and  8  ft.  deep  in  12  days  of  8  hours  each,  in  how 
many  days  of  10  hours  each  can  8  men  excavate  a  cellar  30 
ft.  long,  27  ft.  wide,  and  6  ft.  deep? 

11.  If  5  iron  bars,  4  ft.  long,  3  in.  broad,  and  2  in.  thick, 
weigh  240  lb.,  what  will  be  the  weight  of  20  bars,  each  6  ft. 
long,  21  in.  broad,  and  1 J  in.  thick  ? 

12.  If  9  bricklayers  can  lay  a  wall  80  ft.  long,  20  ft. 
high,  and  1|-  ft.  thick,  in  15  days  of  9  hr.  each,  in  how  many 
days  of  10  hr.  each  can  12  bricklayers  lay  a  wall  100  ft. 
long,  25  ft.  high,  and  2  ft.  thick? 

13.  If  240  men,  in  11  days  of  8  hours  each,  dig  a  ditch 
350  ft.  long,  11  ft.  wide,  and  2J  ft.  deep,  in  how  many  days 
of  9  hours  each  will  48  men  dig  a  ditch  500  ft.  long,  16f  ft. 
wide,  and  3J  ft.  deep  ? 

14.  If  54  men,  in  28  days  of  10  hours  each,  dig  a  trench 
352  yards  long,  2-J-  yards  broad,  and  1J  yards  deep,  how 
long  a  trench  2J  yards  broad,  and  1J  yards  deep,  will  112 
men  dig  in  25  days  of  8J  hours  each  ? 

15.  If  a  regiment  of  1025  soldiers  consumes  11,500  pounds 
of  bread  in  15  days,  how  many  pounds  will  3  regiments  of 
the  same  size  consume  in  12  days  ? 

16.  If  the  water  that  fills  a  vat,  which  is  8  feet  long, 
4  feet  wide,  and  5  feet  deep,  weighs  10,000  pounds,  what 
will  be  the  weight  of  the  water  required  to  fill  a  vat,  which 
is  10  feet  long,  5  feet  wide,  and  6  feet  deep  ? 

17.  If  5  horses  eat  as  much  as  6  cattle,  and  8  horses  and 
12  cattle  eat  12  tons  of  hay  in  40  days,  how  much  hay  will 
be  needed  to  keep  7  horses  and  15  cattle  65  days  ? 


PARTITIVE  PROPORTION.  323 

PARTITIVE  PROPORTION. 

438.  The  process  by  which  a  number  is  divided  into 
parts,  proportional  to  other  given  numbers,  is  called  Parti- 
tive Proportion. 

WRITTEN   EXERCISES. 

439.  1 .   Divide  240  into  parts  proportional  to  3,  4,  and  5. 

EXPLANATION.  —  Since  the  parts  are  proportional  to  3,  4,  and  5, 
out  of  every  12  (the  sum  of  3,  4,  and  5),  there  is  a  3,  a  4,  and  a  5. 
Consequently  one  part  will  be  T\  of  240,  or  60,  another  will  be  T\  of 
240,  or  80,  and  the  other  T\  of  240,  or  100. 

Therefore  the  parts  are  60,  80,  and  100. 

2.  Divide  $  390  into  parts  proportional  to  i,  -J,  and  ^. 

EXPLANATION.  —  Since  fractions  have  the  ratios  of  their  numerators 
when  their  denominators  are  the  same,  the  fractions  are  changed  to 
12ths,  and  we  have  T6j,  T4j,  T3^. 

Therefore  the  problem  may  be  expressed  thus  :  Divide  $  G90  into 
parts  proportional  to  6,  4,  and  3.  This  is  solved  in  the  same  way  as 
example  1. 

3.  Divide  420  into  three  parts  which  shall  be   to  one 
another  as  2,  5,  and  7. 

4.  Divide   750   into   five   parts   which   shall   be  to  one 
another  as  1,  2,  3,  4,  and  5. 

5.  Divide  468  into  three  parts,  such  that  they  shall  be 
proportional  to  1,  -J-,  and  £. 

6.  Divide  $  1596  into  parts  proportional  to  -f,  f ,  and  £ . 

7.  A  man  bought  three    farms   for   $26,150,  and  the 
prices  paid  for  them  were  in  the  proportion  of  the  fractions 
I-,  -J ,  and  -f-.    What  did  he  pay  for  each  farm  ? 

8.  A  man  bequeathed  his  property  in  such  a  way  that 
his  wife  received  $  7  for  every  $  5  received  by  each  of  his 
two  sons  and  every  $  4  received  by  each  of  his  three  daugh- 
ters.    If  his  estate  was  worth  $  250,000,  what  was  the  sum 
bequeathed  to  each  of  the  heirs  ? 


INVOLUTION. 


440.  1.    Of  what  number  are  4  and  4  the  factors  ?   5  and 
5  ?   6  and  6  ?   3,  3,  and  3  f  4,  4,  and  4  ?  5,  5,  and  5  ? 

2.  What  is  the  product  of  6  used  twice  as  a  factor,  or  the 
second  power  of  6  ? 

3.  What  is  the  second  power  of  7  ?   Of  5?   Of  9?   Of  10? 
Of  12? 

4.  What  is  the  third  power  of  2  ?    Of  3  ?    Of  4?    Of  5? 

5.  What  is  the  second  power  of  f  ?   Of  f  ?   Of  £  ?   Of  f  ? 

441.  The  product  arising  from  using  a  number  a  certain 
number  of  times  as  a  factor  is  a  Power  of  the  number. 

442.  The  powers  of  a  number  are  named  from  the  num- 
ber of  times  it  is  used  as  a  factor. 

Thus,  4  is  the  second  power  of  2  ;  9  the  second  power  of  3 ;  8  the 
third  power  of  2  ;  27  the  third  power  of  3. 
The  number  itself  is  called  its  first  power. 

443.  The  number  of  times  a  number  is  used  as  a  factor  is 
indicated  by  a  small  figure,  called  an  Exponent,  written  a 
little  above  and  at  the  right  of  the  number. 

Thus,  32  means  the  second  power  of  3  ;  34  the  fourth  power  of  3,  etc. 

Since  the  area  of  a  square  is  the  product  of  two  equal  factors,  and 
the  volume  of  a  cube  the  product  of  three  equal  factors,  the  second 
power  is  called  the  square,  and  the  third  power  the  cube. 

444.  The  process  of  finding  the  power  of  a  number  is 
called  Involution. 

324 


WRITTEN  EXERCISES.  325 

WRITTEN   EXERCISES. 

445.  1.    Find  the  fourth  power  of  8. 
SOLUTION.  — 8  x  8  x  8  x  8  =  4096,  the  fourth  power  of  8. 

2.  Find  the  second  power  of  13,  18,  21,  36. 

3.  Find  the  third  power  of  9,  15,  24,  42. 

4.  What  is  the  square  of  25  ?   32?   48?   66? 

5.  What  is  the  cube  of  22  ?   25?   54?   71? 

G.  What  is  the  fourth  power  of  4  ?   6  ?   12  ?   19  ? 

7.  What  is  the  third  power  of  23  ?   30?   43?   75? 

8.  What  is  the  square  of  45  ?   69  ?   86  ?   94  ? 

9.  What  is  the  square  off?   f  ?   f  ?   f? 

10.  What  is  the  cube  of  i  ?   f?   f?   f? 
Raise  the  following  to  the  powers  indicated : 

11.  852.  16.    6.52.  21.    (||)2.  26. 

12.  493.  17.    .75?.  22.    (ff)2.  27. 

13.  402.  18.    .053.  23.    (||)2.  28.    (4.Q1)2. 

14.  503.  19.    .0042.         24.    (f)3.  29.    (3.50|)2. 

15.  1022.         20.    3.052.         25.    (if)3.  30.    (5.021)2. 

446.  1.    Find  the  square  of  35  in  terms  of  its  tens  and 
units. 

35  =  30  +  5. 

30+5 
30+5 

302  +  30  X  5 
+  30x5  +  52 


302  +  2(30x5)  +  52 

447.  PRINCIPLE.  —  The  square  of  any  number  consisting 
of  tens  and  units,  is  equal  to  the  tens2  +  2  times  the  tens  x  the 
units  +  the  units2. 


326  INVOLUTION. 

Thus,          25  =  20  +  5,  and  252  =  202  +  2(20  x  5)  +  52. 

The  above  principle  is  true  into  whatever  two  parts  the  number 
may  be  separated,  and  the  principle  stated  in  general  terms  would  be, 
the  square  of  any  number  consisting  of  two  parts  is  equal  to  the  first 
part2  +  2  times  the  first  part  x  the  second  -f  second  part2. 

Thus,          14  =  8  +  6,  and  142  =  82  -f  2  (6  x  8)  +  62. 

Express  in  terms  of  their  tens  and  units  the  square  of 
the  following  numbers : 

2.  54.  5.    47.  8.    74.  11.    39. 

3.  71.  6.    89.  9.    95.  12.    44. 

4.  68.  7.   26.  10.    82.  13.    67. 

448.  1.  Find  the  cube  of  35  in  terms  of  its  tens  and 
units. 

35  =  30  +  5. 

30+5 
30+5 

302  +  30x5 
+  30  x  5  +  5» 


2(30x5) 
30+5 


2(302x5)+    (30  x52) 
+    (302x5)  +  2(30x52)  +  58 

303  +  3(302  x  5)  +  3(30  x  52)  +  53  =  42875. 

449.  PRINCIPLE.  —  TJie  cube  of  a  number  consisting  of  tens 
and  units  is  equal  to  the  tens3  +  3  tens2  x  the  units  +  3  times 
the  tens  x  units2  +  units3. 

Thus,  25  =  20  +  5,  and  258  =  20*  +  3(202  x  5)  +  3(20  x  52)  -f  5s. 

Express  in  terms  of  their  tens  and  units  the  cube  of  the 
following  numbers  : 

2.  27.  5.    43.  8.    46.  11.    66. 

3.  36.  6.   51.  9.    55.  12.    58. 

4.  29.  7.   44.  10.    64.  13.    75. 


EVOLUTION. 


450.  1.  What  are  the  two  equal  factors  of  25?    36?    49? 
2.   What  are  the  three  equal  factors  of  8?  27?  64?  125? 

451.  One  of  the  equal  factors  of  a  number  is  a  Root  of 
the  number. 

Thus,  4  is  a  root  of  16  and  3  is  a  root  of  27. 

452.  Roots  are  named  in  a  manner  similar  to  powers. 

Thus,  one  of  the  two  equal  factors  of  a  number  is  the  second  or 
square  root ;  one  of  the  three  equal  factors,  the  third  or  cube  root ; 
one  of  the  four  equal  factors  the  fourth  root,  etc. 

453.  The  process  of  finding  the   roots   of  numbers   is 
called  Evolution. 

EVOLUTION   BY   FACTORING. 

454.  1.   What  is  the  cube  root  of  4096? 

EXPLANATION.  —  Since  the  cube  root  of  a  number  is  one  of  the  three 
equal  factors  of  it,  the  number  4096  is  separated  into  its  prime  factors, 
and  the  product  of  one  of  the  three  equal  sets  of  prime  factors  is  the  cube 
root.  The  prime  factors  are  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  and  2,  2,  2,  2, 
is  one  of  the  three  equal  sets.  Therefore  2  x  2  x  2  x  2  or  16  is  the 
cube  root  of  4096. 

2.  Find  the  square  root  of  225,  1296,  2401. 

3.  Find  the  square  root  of  11025,  14400,  46656. 

4.  Find  the  cube  root  of  343,  1728,  15625. 

5.  Find  the  cube  root  of  19683,  32768,  74088. 

6.  Find  the  4th  root  of  65536.     The  5th  root  of  248832. 

7.  Find  the  4th  root  of  331776.   The  6th  root  of  2985984. 

327 


328  EVOLUTION. 

SQUARE  ROOT. 

I2  =  1  102  =  100  1002  = 10000 

92  =  81  992  =  9801  9992  =  998001 

455.  1.    How  many  figures  are  required  to  express  the 
square  of  any  number  of  units  ? 

2.  How  does  the  number  of  figures  required  to  express 
the  second  power  of  any  number  between  9  and  100  com- 
pare with  the  number  of  figures  in  the  number  ? 

3.  How  does  the  number  of  figures  expressing  the  second 
power  of  any  number  between  99  and  1000  compare  with 
the  number  of  figures  in  the  number  ? 

4.  If  the  second  power  of  a  number  is  expressed  by  3 
figures,  how  many  orders  of  units  are  there  in  the  number  ? 
If  by  4,  how  many  ?     By  5?     By  7? 

456.  PRINCIPLES.  —  1.    TJie   square  of  a  number  is  ex- 
pressed by  twice  as  many  figures  as  is  the  number  itself,  or  by 
one  less  than  twice  as  many. 

2.  The  orders  of  units  in  the  square  root  of  a  number  corre- 
spond to  the  number  of  periods  of  two  figures  each  into  which 
the  number  can  be  separated,  beginning  at  units. 

The  left-hand  period  may  contain  only  one  figure. 

WRITTEN    EXERCISES. 

457.  1.    What  is  the  square  root  of  576,  or  what  is  the 
side  of  a  square  whose  area  is  576  square  units  ? 

576(20         EXPLANATION.  — According  to  Prin.  2, 

202  =  400     4     Art'  456' the  orders  of  units  in  tne  square 

.  —     root  of  any  number  may  be  determined 

2  X  20  =  40)176  24     by  separating  the  number  into  periods  of 

(40  +  4    x  ^  —  176  two  fig111"613    each,    beginning    at    units. 

Separating  576  thus,  there  are  found  to  be 

two  orders  oi  units  in  the  root,  or  it  is  composed  of  tens  and  units. 


SQUARE   ROOT. 


329 


Since  the  square  of  tens  is  hundreds,  5  hundreds  must  be  the  square 
of  at  least  2  tens.     2  tens  or  20  squared  is  400,  and  400  subtracted 
from  576  leaves  176,  therefore  the  root, 
20,  must  be  increased  by  such  an  amount 
as  will  exhaust  the  remainder. 

The  square  (A)  already  formed  from 
the  576  square  units  is  one  whose  side  is 
20  units,  but  inasmuch  as  the  number  of 
units  was  not  exhausted,  such  additions 
must  be  made  to  the  square  as  will  ex- 
haust the  units  and  keep  the  figure  a 
square.  The  necessary  additions  are  two 
equal  rectangles  B  and  C,  and  a  small 
square  D. 

Since  the  square  D  is  small,  the  area  of 
the  rectangles  B  and  C  is  nearly  176 
units.  The  area,  176  units,  divided  by 
the  length  of  the  rectangles,  will  give  the 
width,  which  is  4  units.  The  width  of 
the  additions  is  4  units,  and  the  entire 
length,  including  the  small  square,  is  44 
units  ;  therefore  the  area  of  all  the  addi- 
tions is  4  times  44  units,  or  176  square 
units,  which  is  equal  to  the  entire  num- 
ber of  units  to  be  added.  Therefore  the  side  of  the  square  is  24  units, 
or  the  square  root  of  the  number  is  24. 


20 


2.    Find  the  square  root  of  2809. 


2809(53 
25 


2  x  50  =  100 

3 

103 


309 


309 


EXPLANATION.  —  Since  the  number  can 
be  separated  into  two  periods,  it  is  evident 
that  the  square  root  of  the  number  is  com- 
posed of  tens  and  units.  The  number  of 
tens  in  the  root  cannot  be  greater  than  5. 
Writing  the  5  tens  in  the  root,  squaring 
and  subtracting  from  2809,  there  is  a  re- 
mainder of  309. 

According  to  Art.  447,  when  the  square  of  the  tens  has  been  subtracted, 
the  remainder  is  composed  of  2  times  the  tens  x  the  units  +  the  square 
of  the  units.  Therefore  309  is  2  times  the  tens  x  the  units + the  square 
of  the  units. 

Since  2  times  the  tens  x  the  units  is  much  more  than  the  square  of 
the  units,  309  is  nearly  two  times  the  tens  x  the  units.  Therefore  309 


330 


EVOLUTION. 


divided  by  two  times  the  tens,  or  100,  is  approximately  the  units  of  the 
root.  Dividing,  the  units  are  found  to  be  3. 

Since  2  times  the  tens  are  to  be  multiplied  by  the  units,  and  the 
units  are  to  be  multiplied  by  the  units,  or  squared,  and  the  results 
added,  to  abridge  the  process  the  units  are  added  to  2  times  the  tens, 
and  the  result  is  multiplied  by  the  units.  Thus  100  +  3  is  multiplied 
by  3,  making  309. 

Therefore  the  square  root  of  2809  is  53. 

Since  any  root  may  be  regarded  as  composed  of  tens  and 
units,  the  above  process  is  of  general  application. 

Thus,  486  =  48  tens  -f  6  units ;  3456  =  345  tens  +  6  units. 

3.    Find  the  square  root  of  785. 

EXPLANATION. — The  first  figure  of 
7-85  1 28.01785  +  the  root  is  2. 

Regarding  the  2  of  the  root  as  tens, 
and  multiplying  by  2,  the  first  partial 
divisor,  40,  is  found.  Dividing  385  by  it, 
the  second  figure  of  the  root  is  found, 
•which  is  8,  and  this  added  to  40  forms 
the  complete  divisor  48. 

Regarding  28,  the  part  of  the  root 
found,  as  tens,  and  multiplying  by  2,  the 
second  partial  divisor,  560,  is  found. 
Dividing  100  by  it,  the  next  figure  of  the 
root  is  found,  which  is  0,  and  this  added 
to  560  gives  560  for  the  complete  divisor. 
The  other  partial  and  complete  divisors 
are  found  in  the  same  way. 

After  a  number  of  decimal  places  in 
the  root  has  been  found,  a  few  more  may 
be  found  by  ordinary  division,  as  shown 
in  the  example. 

RULE.  —  Separate  the  number  into  periods  of  two  figures 
each,  beginning  at  units. 

Find  the  greatest  square  in  the  left-hand  period,  and  write 
its  root  for  the  first  figure  of  the  required  root. 

Square  this  root  and  subtract  the  result  from  the  left-hand 
period,  and  annex  to  the  remainder  the  next  period  for  a, 
dividend. 


48 


560 


5601 


56027 


4 

385 

384 
100 
000 


10000 
5601 


439900 
392189 


477110 
448216 


288940 
280135 


SQUARE   ROOT.  331 

Double  the  root  already  found  for  a  partial  divisor,  and  by  it 
divide  the  dividend,  disregarding  the  right-hand  figure.  The  quo- 
tient or  quotient  diminished  will  be  the  second  Jigure  of  the  root. 

Annex  to  the  partial  divisor  for  a  complete  divisor  the  Jigure 
last  found,  multiply  this  divisor  by  the  last  figure  of  the  root 
found,  subtract  the  product  from  the  dividend,  and  to  the 
remainder  annex  the  next  period  for  the  next  dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used 
thus.  The  result  will  be  the  square  root  sought. 

1.  When  the  number  is  not  a  perfect  square,  annex  periods  of 
decimal  ciphers  and  continue  the  process. 

2.  Decimals  are  pointed  off  into  periods  of  two  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 

3.  The  square  root  of  a  common  fraction  is  found  by  extracting 
the  square  root  of  both  numerator  and  denominator  separately,  or  by 
reducing  it  to  a  decimal  and  then  extracting  its  root. 

Extract  the  square  root  of  the  following : 

4.  3025.  8.  13225.  12.  41616.  16.  1900.96. 

5.  5476.  9.  14641.  13.  52441.  17.  .514089. 

6.  9604.  10.  21025.  14.  77284.  18.  97.8121. 

7.  11881.  11.  23409.  15.  173056.  19.  .001225. 

20.  89.661961.  23.    JJJf  26.    2450^. 

21.  2540.664025.          24.   ^%rir-  27. 

22.  282429536481.       25.    \H\\\.          28. 

Find  the  square  root  to  five  decimal  places  : 

29.  18.  33.    127.  37.    7.25.  41.  12.6|. 

30.  35.  34.    245.  38.    .526.  42.  25.0f. 

31.  47.  35.    370.  39.    .031.  43.  .004|. 

32.  91.  36.    813.  40.    42.9.  44.  .OOOf. 


332 


EVOLUTION. 


APPLICATIONS  OP  SQUARE  ROOT. 

458.  Since  the  area  of  a  square  is  the  product  of  the  two 
equal  factors  which  represent  its  sides,  it  is  evident  that  the 
length  of  a  side  may  be  found  by  finding  the  square  root  of 
the  area. 

1.  A  man  owns  a  farm  in  the  form  of  a  square  which 
contains  45  A.  25  sq.  rd.     How  many  rods  in  length  or 
breadth  is  it  ? 

2.  What  are  the  dimensions  of  a  rectangular  farm  con- 
taining 80  acres,  whose  length  is  twice  its  breadth  ? 

3.  A  general,  attempting  to  draw  his  army  of  8000  men 
into  a  square,  found  he  had  256  men  over.     What  was  the 
number  of  men  in  rank  and  file  ? 

4.  A  man,  having  a  garden  324  yards  square,  extended 
it  so   as  to  make  it  9  times  as  large.     How  many  yards 
square  was  it  then  ? 

5.  How  much  more  will  it  cost,  at  $1.35  a  rod,  to  fence 
a  field  in  the  form  of  a  rectangle,  135  rods  long  and  60  rods 
wide,  than  to  fence  a  field  of  equal  area  in  the  form  of  a 
square  ? 

459.  Since  the  square  described  upon  the  hypotenuse,  or 
side  opposite  the  right  angle,  of  a 

right-angled  triangle  is  equivalent 
to  the  sum  of  the  squares  upon 
the  other  two  sides,  it  is  evident : 

1st,  That  the  hypotenuse  is  equal 
to  the  square  root  of  the  sum  of  the 
squares  of  the  other  two  sides. 

2d,  That  the  base  or  perpendicular 
is  equal  to  the  square  root  of  the 
difference  of  the  square  of  the  hy- 
potenuse and  that  of  the  other  side. 


APPLICATIONS  OF  SQUARE  ROOT.  333 

6.  A  rope  attached  to  the  top  of  a  derrick  60  feet  high, 
and  drawn  perfectly  straight,  reaches  the  ground  80  feet 
from  the  derrick.     How  long  is  the  rope  ? 

SOLUTION.  —  The  sides  about  the  right  angle  are  60  ft.  and  80  ft. 
respectively,  and  we  are  to  find  the  hypotenuse.  By  Art.  459,  the 
hypotenuse  is  equal  to  the  square  root  of  the  sum  of  the  squares  of  the 
other  two  sides. 

.•.  V602  +  802  =  100,  feet  in  length  of  the  rope. 

7.  Two  rafters,  each  24  feet  long,  meet  at  the  ridge  of 
a  roof  12  feet  above  the  body  of  the  house.     How  wide  is 
the  house  ? 

8.  If  a  line  150  feet  long  will  reach  from  the  top  of  a 
fort  to  the  opposite  side  of  a  river  that  is  85  feet  wide, 
what  is  the  height  of  the  fort  ? 

9.  At  $  1.75  a  rod,  what  will  it  cost  to  fence  a  triangular 
lot,  one  side  of  which,  40  rods  in  length,  forms  a  right 
angle  with  another  side  30  rods  in  length  ? 

10.  How  far  apart  are  the  opposite  corners  of  a  square 
farm  which  contains  360  acres  ? 

11.  A  tree,  broken  off  21  feet  from  the  ground,  and  rest- 
ing on  the  stump,  touches  the  level  ground  28  feet  from  the 
base  of  the  stump.     What  was  the  height  of  the  tree  ? 

12.  What  is  the  distance  from  a  lower   corner  to  the 
opposite  upper  corner  of  a  room  24  feet  long,  18  feet  wide, 
and  12  feet  high  ? 

13.  There  are  two  columns  in  the  ruins  of  Persepolis  left 
standing  upright ;  one  is  70  feet  above  the  plane,  and  the 
other  50  feet  above.     In  a  straight  line  between  these  stands 
a  small  statue,  5  feet  in  height,  the  head  of  which  is  100  feet 
from  the  summit  of  the  higher,  and  80  feet  from  the  top  of 
the  lower  column.     What  is  the  distance  between  the  two 
columns  ? 


334  EVOLUTION. 

SIMILAR  SURFACES. 

460.  Figures  which  have  the  same  form  and  differ  only 
in  size  are  Similar  Figures. 

The  following  truths  regarding  similar  figures  can  be 
established  by  geometry : 

461.  PRINCIPLES.  —  1.    Similar  surfaces  are  to  each  other 
as  the  squares  of  their  corresponding  dimensions. 

2.  The  corresponding  dimensions  of  similar  surfaces  are  to 
each  other  as  the  square  roots  of  their  areas. 

1.  There  are  two  circular  gardens,  one  having  a  diam- 
eter of  8  rods  and  the  other  32  rods.     How  do  they  compare 
in  size  ? 

SOLUTION.  —  Since  the  gardens  are  similar  in  form,  their  sizes  or 
areas  are  to  each  other  as  the  squares  of  the  same  dimensions  in  each  ; 
that  is,  as  82  is  to  322,  or  as  64  is  to  1024.  Since  1024  is  16  times  64, 
the  larger  garden  is,  therefore,  16  times  as  large  as  the  smaller. 

2.  A  lady  had  a  circular  flower-bed  8  feet  in  diameter, 
and  a  similar  one  four  times  as  large.     What  was  the  diam- 
eter of  the  larger  bed  ? 

3.  A  has  a  rectangular  field  80  rods  long  and  60  rods 
wide.     What  will  be  the  dimensions  of  a  similar  field  con- 
taining 13J  acres  ? 

4.  If  a  horse  tied  to  a  stake  by  a  rope  7.13  rods  in  length 
can  graze  upon  just  one  acre  of  ground,  how  long  should  the 
rope  be  to  allow  him  to  graze  upon  6-J-  acres  ? 

5.  If  6  gallons  of  water  flow  through  a  pipe  1  inch  in 
diameter  in  a  minute,  how  many  gallons  will  flow  through 
a  pipe  4  inches  in  diameter  in  5  minutes,  when  the  stream, 
moves  with  the  same  velocity  ? 

6.  A  school-room  contained  two  square  blackboards  whose 
sides  were  3  ft.  and  6  ft.  respectively.     How  did  they  com- 
pare in  area  ? 


CUBE   ROOT.  335 

CUBE  ROOT. 

l'  =  l  10s  =  1000  1003  =  1000000 

33  =  27  36s  =  46656  3613  =  47045881 

93  =  729  99s  =  970299  999s  =  997002999 

462.  1.    How  many  figures  are  required,  to  express  the 
cube  of  any  number  of  units? 

2.  How  does  the  number  of  figures  required  to  express 
the  cube  of  any  number  between  9  and  100  compare  with 
the  number  of  figures  expressing  the  number  ? 

3.  How  does  the  number  of  figures  expressing  the  cube 
of  any  number  between  99  and  1000  compare  with  the  num- 
ber of  figures  expressing  the  number  ? 

4.  If,  then,  the  cube  of  a  number  is  expressed  by  4  figures, 
how  many  orders  of  units  are  there  in  the  root  ?     If  by  5 
figures,  how  many  ?     If  by  6  figures,  how  many  ?     If  by  8 
figures,  how  many  ? 

5.  How  may  the  number  of  figures  in  the  cube  root  of  a 
number  be  found  ? 

463.  PRINCIPLES.  —  1.    The  cube  of  a  number  is  expressed 
by  three  times  as  many  figures  as  the  number  itself,  or  by  one 
or  two  less  than  three  times  as  many. 

2.  TJie  orders  of  units  in  the  cube  root  of  a  number  corre- 
spond to  the  number  of  periods  of  three  figures  each  into  which 
the  number  can  be  separated,  beginning  at  units. 

The  left-hand  period  may  contain  one,  two,  or  three  figures. 

464.  If  the  tens  of  a  number  are  represented  by  t  and  the 
units  by  u,  the  cube  of  a  number  consisting  of  tens  and  units 
will  be  the  cube  of  (t  +  u)  or  1?  +  3  t*u  +  3  tu2  +  u3,  Art.  449. 

Thus,  35  =  3  tens  +  5  units,  or  30  +  5,  and  35s  =  30s  -f  3(302  x  6) 
+  3(30  x  52)  +  53  =  42875. 


336 


EVOLUTION. 


WRITTEN   EXERCISES. 

465.    1.    What  is  the  cube  root  of  13824,  or  what  is  the 
edge  of  a  cube  whose  solid  contents  are  13824  units  ? 

13-824  (20  +  4  =  24         EXPLANATION. — Accord- 
g  QQQ  ing  to  Prin.  2,  Art.  463,  the 


203= 

3x202=1200 

3x4x20=  240 

42=     16 

1456 


5824 


5824 


orders  of  units  in  the  cube 
root  of  any  number  may  be 
determined  from  the  num- 
ber of  periods  obtained  by 
separating  the  number  in- 
to periods  containing  three 


figures  each,  beginning  at  units.     Separating  the  given  number  thus, 
there  are  two  periods,  or  the  root  is  composed  of  tens  and  units. 

The  tens  in  the  cube  root  of 
the  number  cannot  be  greater 
than  2,  for  the  cube  of  3  tens  is 
27000.  2  tens,  or  20  cubed,  are 
8000,  which,  subtracted  from 
13824,  leave  5824;  therefore  the 
root,  20,  must  be  increased  by  a 
number  such  that  the  additions 
will  exhaust  the  remainder. 

The  cube  (A)  already  formed 
from  the  13824  cubic  units  is  one 
whose  edge  is  20  units.  The  ad- 
ditions to  be  made,  keeping  the 
figure  formed  a  perfect  cube,  are 
3  equal  rectangular  solids,  B,  C, 
and  D  ;  3  other  equal  rectangular 
solids,  E,  F,  and  G ;  and  a  small 
cube,  H.  Inasmuch  as  the  solids, 
B,  C,  and  D,  comprise  much  the 
greatest  part  of  the  additions, 
their  solid  contents  will  be  nearly 
5824  cubic  units,  the  contents  of 
the  addition. 

Since  the  cubical  contents  of  these  three  equal  solids  are  nearly 
equal  to  6824  units,  and  the  superficial  contents  of  a  side  of  each  of 
these  solids  are  20  x  20,  or  400  square  units,  if  we  divide  5824  by  3 
times  400,  or  1200,  since  there  are  3  equal  solids,  we  shall  obtain  the 
thickness  of  the  addition,  which  is  4  units. 


CUBE   ROOT. 


337 


Since  all  the  additions  have  the  same  thickness,  if  their  superficial 
contents,  or  area  of  each  side,  are  multiplied  by  4,  the  result  will 
be  the  solid  contents  of  these 
additions. 

Besides  the  larger  additions 
tnere  are  three  others,  E,  F, 
and  G,  which  are  each  20  units 
long  and  4  units  wide,,  or  whose 
surfaces  have  an  area  of  80  units 
each,  or  240  units  altogether; 
and  a  small  cube  whose  sides 
have  an  area  of  16  units.  The 
sum  of  these  areas,  1456,  mul- 
tiplied by  4,  the  thickness  of 
the  additions,  gives  the  solid 
contents  of  the  additions,  which 
are  5824  units. 

Therefore  the  edge  of  the 
cube  is  24  units  in  length,  or 
the  cube  root  of  13824  is  24. 

EXPLANATION.  —  In  the  same 
manner  as  before,  it  may  be 
shown  that  the  root  of  the  num- 
ber contains  only  tens  and  units. 
The  tens  cannot  be  greater  than  2,  for  3  tens  raised  to  the  third  power 
is  27000.  Cubing  2  tens  and  subtracting,  there  is  left  5824. 

This  remainder  contains  3  times  the  tens2  x  the  units  +  3  times 
13-824(24  thetensx  the  units2,  +  the 


*3=203= 

3£2=202x3=1200 

(20x4)  x3=  240 

w2=4x4=     16 


8000 


(3t2+3tu+u2)  xu= 


5824 


5824 


Each  of  these  parts  con- 
tains the  units  as  a  factor, 
hence  5824  is  the  product 
of  two  factors,  one  of  which 
is  the  units,  and  the  other 
3  times  the  tens2  +  3  times 
the  tens  x  the  units  +  the 
units2. 


Since  3  times  the  tens2  is  much  greater  than  the  rest  of  the  factor,  if 
6824  is  divided  by  3  times  the  tens2,  or  1200,  the  quotient  will  be  the 
units  or  other  factor.  It  is  found  to  be  4. 

The  factor  completed  is  therefore  3  x  202  +  3  x  20  x  4  +  42,  which 
is  equal  to  1200  +  240  +  16,  or  1456.  This  multiplied  by  4  gives  the 
product  5824.  Therefore  the  cube  root  of  the  number  is  24. 

STAND.    AR. 22 


338 


EVOLUTION. 


When  the  number  consists  of  more  than  two  orders  of 
units,  the  root  may  be  found  in  the  same  manner  by  consid- 
ering each  time  the  root  already  found  as  tens  and  the  next 
evrder  of  the  root  as  units. 


2.    What  is  the  cube  root  of  48228544  ? 


Partial  divisor,    3  x  302  =  2700 

3x30x6=  540 

62  =  36 

Complete  divisor,  3276 

Partial  divisor,  3  x  3602  =  388800 
3  x  360  x  4  =     4320 

42= 16 

Complete  divisor,  393136 


48-228-544  [  364 

27 


21228 


19656 


1572544 


1572544 


3.   What  is  the  cube  root  of  22906304? 


22-906-304  |_284 
8 


3  x  202  =  1200 
3x20x8=   480 

82=i     64 
1744 


3  x2803 
3  X  280  x  4 


235200 

3360 

_  16 

238576 


14906 


13952 


954304 


954304 


When  the  number  of  figures  in  the  root  is  more  than  two 
the  following  method  materially  abridges  the  process  : 


CUBE  ROOT. 


339 


4.   What  is  the  cube  root  of  4  to  4  decimal  places? 

411.5874 


u2  = 


3xl02=300 
3  x  10  x  5  =  150 
52=_25) 
475V 
175) 

3x1502  =  67500 
3  x  150  x  8  =   3600 

82  = 64) 

71164  [. 
3664) 

3x15802=7489200 
3  x  1580  x  7  =     33180 

72= 49) 

7522429  [• 
33229  ) 


3  x  158702  =  755570700 


3000 


2375 


625000 


569312 


55688000 


52657003 


3030997000 


3022282800 


EXPLANATION. — Since  the  root  of  the  number  is  not  a  whole  num- 
ber, periods  of  decimal  ciphers  are  annexed,  and  the  required  number 
of  decimal  places  found. 

After  two  figures  of  the  root  have  been  found,  the  partial  divisors 
may  be  found  as  follows :  Add  together  3  times  the  product  of  the  tens 
by  the  units  and  the  square  of  the  units,  then  add  this  sum  to  the 
complete  divisor,  plus  the  square  of  the  units,  and  the  result,  with  two 
ciphers  annexed,  will  be  the  next  partial  divisor. 

Thus,  in  the  example  solved,  to  obtain  the  partial  divisor  for  the 
third  figure  of  the  root  add  150  and  25,  and  place  their  sum  immediately 
below  the  complete  divisor.  Then  add  together  175,  475,  and  25,  and 
to  the  sum  annex  two  ciphers.  The  result  will  be  the  next  partial 
divisor. 

After  several  decimal  places  have  been  found  a  few  more  may  be 
found  by  ordinary  division. 

It  will  be  seen  by  examining  the  solution  that  the  numbers  added 
together  are  equal  to  3 12  +  6  tu  -f  3  w2,  or  3  (£2  +  2  tu  -f  w2),  that  is,  the 
sum  is  3  times  the  square  of  the  tens  and  units  already  found. 


340  EVOLUTION. 

RULE.  — Separate  the  number  into  periods  of  three  figures 
each,  beginning  at  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  write  its 
root  for  the  first  figure  of  the  required  root.. 

Cube  this  root,  subtract  the  result  from  the  left-hand  period, 
and  annex  to  the  remainder  the  next  period  for  a  dividend. 

Take  three  times  the  square  of  the  root  already  found,  con- 
sidered as  tens,  for  a  partial  divisor,  and  by  it  divide  the 
dividend.  The  quotient  or  the  quotient  diminished  will  be  the 
second  part  of  the  root. 

To  this  partial  divisor  add  three  times  the  product  of  the 
first  part  of  the  root,  considered  as  tens,  by  the  second  part,  and 
also  the  square  of  the  second  part.  Their  sum  will  be  the  com- 
plete divisor. 

Multiply  the  complete  divisor  by  the  second  part  of  the  root, 
and  subtract  the  product  from  the  dividend. 

Continue  thus  until  all  the  figures  of  the  root  have  been  found. 

1.  When  there  is  a  remainder,  after  subtracting  the  last  product 
annex  periods  of  decimal  ciphers,  and  continue  the  process.    The 
figures  of  the  root  obtained  after  the  ciphers  are  annexed  will  be 
decimals. 

2.  Decimals  are  pointed  off  into  periods  of  three  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 

3.  The  cube  root  of  a  common  fraction  is  found  by  extracting  the 
cube  root  of  both  numerator  and  denominator  separately,  or  by  reduc- 
ing it  to  a  decimal  and  then  extracting  its  root. 

Extract  the  cube  root  of  the  following : 

5.  54872.  10.   43614208.  15.   491916472984 

6.  175616.         11.   130323843.  16.   13312.053. 

7.  405224.         12.   1865409391.          17.   28.094464. 

8.  857375.         13.   4065356736.          18.   .000166375 

9.  3048625.       14.   95256152263.        19.   .000001953125. 

20.  What  is  the  cube  root  of  2  to  four  decimal  places  ? 

21.  What  is  the  cube  root  of  6  to  five  decimal  places? 

22.  What  is  the  cube  root  of  $?  Jfffa?  Y^Hf**? 


APPLICATIONS  OF   CUBE   ROOT.  341 

APPLICATIONS  OF  CUBE  ROOT. 

466.  Since  the  volume  of  a  cube  is  the  product  of  the- 
three  equal  factors  that  represent  its  edges,  it  is  evident  that 
the  cube  root  of  the  volume  gives  the  length  of  the  edge. 

1.  A  cubical  box  contains  54,872  cubic  inches.     What  is 
the  length  of  each  side  ? 

2.  How  deep  is  a  cubical  cistern  containing  2744  cu.  ft.  ? 

3.  What  is  the  number  of  square  inches  in  one  face  of  a. 
cubical  block  whose  contents  are  185,193  cubic  inches  ? 

4.  What  are  the  dimensions  of  a  cubical  box  which  con- 
tains as  much  as  a  rectangular  box  5  ft.  4  in.  long,  4  ft.  6  in. 
wide,  and  2  ft.  8  in.  deep  ? 

5.  What  must  be  the  depth  of  a  cubical  bin  that  will 
contain  exactly  1200  bushels  ? 

6.  A  cubical  cistern  holds  400  barrels  of  water.     How 
deep  is  it  ? 

7.  How  much  will  it  cost,  at  35  cents  per  square  yard,  to- 
plaster  the  bottom  and  sides  of  the  cistern  ? 

8.  A  bin  that  is  just  twice  as  long  as  it  is  wide  or  high 
holds  500  bushels  of  grain.     What  is  its  length  ? 

9.  Which  has  the  greater  surface  and  how  much ;  a  cube- 
whose  solid  contents  are  4096  cubic  feet,  or  a  rectangular 
solid  having  the  same  contents,  whose  width  is  twice  its, 
height,  and  whose  height  is  one  third  its  length  ? 

SIMILAR  VOLUMES. 

The  following  principles  are  proved  by  geometry : 

467.  PRINCIPLES.  —  1.   Similar  solids  are  to  each  other  as 
the  cubes  of  their  like  dimensions.     Hence, 

2.    The  corresponding  dimensions  of  similar  solids  are  to- 
each  other  as  the  cube  roots  of  their  volumes. 


342  EVOLUTION. 

1.  If  a  globe  4  inches  in  diameter  weighs  8  lb.,  what  will 
be  the  diameter  of  a  similar  one  that  weighs  125  lb.  ? 

EXPLANATION. — Since  the  correspond- 

4  •  x  ' '  "v/8  *  -v/125    Cl")      *n£  dimensions  of  similar  solids  are  pro- 

portional  to  the  cube  roots  of  these  vol- 

4:  a;::  J:  o  (Z)  Umes,  we  have  the  diameter  of  the  smaller 
X  =  10,  inches  in  diam.  globe  4  inches  :  the  diameter  of  the 

larger  globe  x  ::  the  cube  root  of  the 

weight  of  the  smaller  globe  VS  :  the  cube  root  of  the  weight  o} 
the  other  globe  \/125  (1).  Extracting  the  cube  root  of  8  and  125,  we 
have  (2).  Whence,  solving,  the  diameter  is  10  inches. 

2.  There  are  two  cubes  whose  dimensions  are  4  inches 
and  16  inches  respectively.     The  larger  is  how  many  times 
the  smaller  ? 

3.  If  a  ball  3  inches  in  diameter  weighs  7  pounds,  what 
will  be  the  weight  of  a  similar  ball  5  inches  in  diameter  ? 

4.  A  cubical  bin  5  feet  long  will  hold  100.44  bushels. 
How  much  will  a  cubical  bin  10  feet  long  hold  ? 

5.  The  height  of  a  cubical  vessel  is  1  foot  6  inches.    How 
high  must  another  cubical  vessel  be  to  hold  four  times  as 
much? 

6.  If  a  globe  of  gold  1  inch  in  diameter  is  worth  $  120, 
what  is  the  diameter  of  a  globe  of  gold  worth  $  6400  ? 

7.  If  a  man  5  ft.  6  in.  high  weighs  140  pounds,  what  is 
the  weight  of  a  man  of  similar  build  whose  height  is  6  ft.  ? 

8.  There  are  two  balls  whose  diameters  are  4  inches  and 

5  inches  respectively.     What  is  the  diameter  of  a  ball  whose 
contents  are  equal  to  them  both  ? 

9.  If  a  haystack  13  feet  in  diameter  contains  17  tons, 
what  is  the  diameter  of  a  similar  stack  which  contains  136 
tons? 

10.  A  bushel  measure  is  in  the  form  of  a  cylinder  18^-  in. 
in  diameter,  and  8  in.  deep.  What  will  be  the  dimensions 
of  a  peck  measure  of  similar  shape  ? 


GENERAL   REVIEW   EXERCISES. 


ORAL  EXERCISES. 

468.     1.    Two  boys  have  together  45  cents,  but  one  has 
twice  as  much  as  the  other.     How  many  cents  has  each  ? 

2.  If  a  man  can  do  -|  of  a  piece  of  work  in  a  day,  how 
long  will  it  take  him  to  do  one  half  of  it  ? 

3.  If  5  men  can  do  a  piece  of  work  in  12  days,  how  long 
will  it  take  8  men  to  do  it  ? 

4.  A  man  bought  sheep  at  $  3  a  head.     Had  he  paid  $  5 
a  head  they  would  hare  cost  $  16  more.     How  many  did  he 
buy? 

5.  In  what  time  can  40  men  do  a  piece  of  work  that  50 
men  can  do  in  8  days  ? 

6.  A  can  do  a  piece  of  work  in  3  days  and  B  in  41  days. 
In  what  time  can  they  together  do  it  ? 

7.  James  and  Henry  can  hoe  a  field  in  5  days.     James 
can  do  it  alone  in  9  days.     In  how  many  days  can  Henry 
hoe  the  field  alone  ? 

8.  A  can  make  a  door  in  -|  of  a  day,  and  B  in  f  of  a  day. 
How  many  doors  can  they  together  make  in  a  day  ? 

9.  How  long  will  it  take  A  to  finish  a  door  after  B  has 
worked  on  it  half  a  day  ? 

10.  A,  B,  and  C  can  do  a  piece  of  work  in  4  days.     A  can 
do  it  alone  in  12  days,  and  B  alone  in  15  days.     How  long 
will  it  take  C  to  do  it  alone  ? 

11.  If  I  lose  fy  of  my  money,  and  spend  -J  of  the  remain- 
der, what  part  have  I  left  ? 

343 


344  GENERAL   REVIEW  EXERCISES. 

12.  Three  boys,  Peter,  George,  and  Jacob,  can  do  a  piece 
of  work  in  3  days.     Peter  can  do  it  alone  in  12  days,  and 
Peter  and  Jacob  can  do  it  in  8  days.     How  long  will  it  take 
each  of  them  to  do  it  ? 

13.  If  I  gain  ^  of  a  cent  apiece  by  selling  eggs  at  8  cents 
a  dozen,  how  much  apiece  will  I  gain  by  selling  them  at  10 
cents  a  dozen  ? 

14.  If  I  sell  my  apples  at  6  cents  a  dozen,  I  lose  15  cents ; 
but  if  I  sell  them  at  9  cents  a  dozen,  I  gain  12  cents.     How 
many  have  I,  and  what  did  they  cost  me  ? 

15.  If  -|  of  A's  money  equals  |-  of  B's,  what  part  of  B's 
equals  |-  of  A's  ? 

16.  Five  times  T^  of  a  number  is  14  less  than  the  number. 
"What  is  the  number  ? 

17.  I  sold  a  bureau  to  A  for  -|-  more  than  it  cost  me.     He 
sold  it  for  $  6,  which  was  -§•  less  than  it  cost  him.     What 
did  it  cost  me  ? 

18.  A  man  agreed  to  work  16  days  for  $  24  and  board, 
but  he  was  to  pay  $  1  a  day  for  his  board  for  every  day  he 
was  idle.     He  received  $  14.     How  many  days  did  he  work  ? 

19.  B  engaged  to  work  20  days  for  $40,  and  agreed  to 
forfeit  $!£  for  every  day  he  was  idle.     How  many  days 
ivas  he  idle,  if  he  received  $  29^-  ? 

20.  Two  persons  share  $150  in  the  ratio  of  \  and  ^. 
"What  is  the  share  of  each  ? 

21.  A  and  B  engaged  in  a  business  in  which  A  invested 
$  36  and  B  received  $  5  out  of  the  $  8  which  they  gained. 
How  much  did  B  invest  ? 

22.  Three  persons  are  to  share  a  certain  sum  of  money 
in  the  ratio  of  -g>  ^,  and  £.     The  second  receives  $  9  more 
than  the  third.     What  is  the  share  of  each  ? 

23.  If  a  man  can  earn  -|  of  a  dollar  in  %  of  a  day,  how 
much  can  he  earn  in  -|  of  a  day  ? 


ORAL  EXERCISES.  345 

24.  If  f  of  the  value  of  a  carriage  is  equal  to  J  of  the 
value  of  a  horse,  and  the  value  of  the  carriage  is  $  20  more 
than  the  value  of  the  horse,  what  is  the  value  of  each  ? 

25.  In  an  orchard  J  of  the  trees  bear  apples,  ^  bear  pears, 
and  the  remainder,  300,  bear  peaches.     How  many  trees 
are  there  in  the  orchard  ? 

26.  A  man  can  saw  2  cords  of  wood  per  day,  or  he  can 
split  3  cords  of  wood  when  sawed.     How  much  must  he  saw 
'that  he  may  be  occupied  the  rest  of  the  day  in  splitting  it  ? 

27.  A  man  spent  one  half  of  his  money  and  half  a  dollar 
for  a  coat,  one  half  of  what  he  had  left  and  half  a  dollar  for 
a  hat,  one  half  of  what  was  left  and  half  a  dollar  for  shoes, 
and  had  a  dollar  left.     How  much  had  he  at  first  ? 

28.  A  merchant,  after  selling  from  a  cask  of  vinegar  15 
gallons  more  than  ^  of  the  whole,  found  that  he  had  left  just 
4  times  as  much  as  he  had  sold.     How  many  gallons  did  the 
cask  contain  at  first  ? 

29.  Three  boys  had  together  earned  150  cents.     James 
had  earned  ^  as  much  as  John,  and  Henry  ^  as  much  as 
Jaines  and  John.     What  sum  had  each  earned  ? 

30.  A  tree  129  feet  high  was  broken  in  a  storm.     -|  of  the 
part  broken  off  was  equal  to  -f-  of  the  part  standing.     What 
was  the  length  of  each  part  ? 

31.  Wheat  sold  at  $1.50  per  bushel  pays  a  profit  of  one 
half  the  cost.     If  it  is  sold  at  $  2  per  bushel,  what  part  of 
the  cost  will  be  gained  ? 

32.  If  a  merchant  sells  f  of  an  article  for  what  J-  of  it 
cost,  what  is  his  gain  per  cent  ? 

33.  I  sold  some  goods  at  a  discount  of  40%,  and  10%  off 
for  cash.     What  was  the  total  %  discount  ? 

34.  If  goods  are  bought  at  80%  discount,  and  20%  off  for 
cash,  what  is  the  entire  %  discount  ? 


346  GENERAL  REVIEW  EXERCISES. 

35.  Which,  is  better,  and  how  much,  to  buy  goods  at  25% 
discount  and  10%  off  for  cash,  or  to  buy  goods  at  10%  dis- 
count and  25%  off  for  cash  ? 

36.  A  wholesale  merchant  offered  dress  goods,  marked 
at  50  cents  per  yard,  at  a  discount  of  20%,  and  5%  off  for 
cash.     At  what  price  per  yard  did  he  offer  them  ? 

37.  A  man  purchased  a  horse,  giving  in  payment  his  note 
at  6%.     At  the  end  of  3  years  and  6  months  he  found  that 
he  owed  $  42  interest.     How  much  did  the  horse  cost  him  ? 

38.  A  man  wished  to  invest  enough  money  in  govern- 
ment  bonds   paying  4%    interest  annually  to   secure   an 
income  of   $800.     How  many   one-thousand-dollar   bonds 
must  he  purchase  ? 

39.  What  principal  will  amount  to  $1300  in  6  years, 
with  interest  at  5  %  ? 

40.  What  principal  will  amount  to  $850  in  10  years, 
with  interest  at  7%? 

41.  If  I  had  paid  8%  less  for  my  house  than  I  did,  it 
would  have  made  a  difference  of  $  400  in  the  cost.     What 
did  it  cost  me  ? 

42.  A  and  B  each  had  farms  containing  240  acres.     A 
purchased  a  certain  number  of  acres  from  B  and  he  then 
had  320  acres.     What  per  cent  of  A's  farm  was  B's  after 
the  purchase? 

43.  The  retail  price  of  some  books  was  $  1  per  volume. 
If  I  bought  them  at  a  discount  of  20%  from  the  retail  price, 
and  sold  them  at  the  retail  price,  what  per  cent  did  I  gain  ? 

44.  If  goods  are  sold  so  that  -f-  of  the  cost  is  received  for 
half  the  quantity  of  goods,  what  is  the  gain  per  cent  ? 

45.  I  asked  20%  more  for  goods  than  they  cost  me,  but 
sold  them  at  10%  less  than  I  asked  for  them.     What  per 
cent  did  I  gain? 


ORAL   EXERCISES.  347 

46.  A  sells  a  horse  to  B,  gaining  20%,  and  B  sells  it  to 
C  for  f  150,  and  gains  25%.     What  did  the  horse  cost  A  ? 

47.  If  a  banker  sells  a  sight  draft  on  New  Orleans  for 
$5000  for  $5012.50,  what  is  the  rate  of  exchange? 

48.  A  pole  which  is  10  feet  long  casts  a  shadow  4  feet 
long,  and  at  the  same  time  the  shadow  of  a  steeple  is  30 
feet  long.     How  high  is  the  steeple  ? 

49.  Six  times  a  number  equals  5  times  -J  of  the  same 
number,  plus  33.     What  is  the  number  ? 

50.  A  man,  being  asked  how  many  sheep  he  had,  replied, 
"  If  I  had  3  times  as  many  as  I  have  and  5  sheep,  I  would 
have  185."     How  many  had  he  ? 

51.  What  time  after  2  o'clock  are  the  hour  and  minute 
hands  of  a  clock  together  ? 

52.  A  person,  being  asked  the  time  of  day,  replied  that 
it  was  past  noon,  and  that  f  of  the  time  past  noon  was 
equal  to  -|  of  the  time  to  midnight.     What  was  the  time  ? 

53.  A,  B,  and  C  ate  8  loaves  of  bread.     A  furnished  3 
loaves  and  B  5  loaves.     C  paid  the  others  24  cents  for  his 
share.     How  much  money  should  A  and  B  each  receive  ? 

54.  How  far  may  a  person  ride  in  a  stage,  going  at  the 
rate  of  8  miles  an  hour,  if  he  is  gone  11  hours,  and  walks 
back  at  the  rate  of  3  miles  an  hour  ? 

55.  A  yacht,  whose  rate  of  sailing  is  12  miles  an  hour, 
sails  down  a  river  whose  current  is  4  miles  an  hour.     How 
far  may  it  go,  if  it  is  to  be  gone  15  hours  ? 

56.  A  steamboat  sailed  42J  miles  in  2-J-  hours.     How  far 
did  it  sail  in  20  minutes  ? 

57.  If  7  men  can  dig  32  rods  of  ditch  in  1  day,  how  many 
men  will  be  required  to  dig  96  rods  in  f  of  a  day  ? 

58.  I  have  pasturage  for  either  12  horses  or  18  cows  on 
my  farm.     If  I  have  6  cows,  how  many  horses  can  I  keep  ? 


348  GENERAL   REVIEW   EXERCISES. 

59.  If  it  costs  $  50  to  support  8  persons  for  2^  weeks, 
what  will  it  cost  to  support  10  persons  for  3  weeks  ? 

60.  A  is  25  years  old  and  B  is  4  years  old.    In  how  many 
years  will  A  be  four  times  as  old  as  B  ? 

61.  John  is  20  years  old,  which  is  J  of  his  uncle's  age. 
How  long  since  his  uncle  was  twice  as  old  as  John  ? 

62.  Mr.  A.  is  35  years  of  age  and  his  son  is  10.     How 
soon  will  the  son  be  one  half  the  age  of  the  father  ? 

63.  Ten  years  ago  C  was  -J  as  old  as  D,  but  now  he  is 
^  as  old.     What  is  the  age  of  each? 

64.  A  farmer  one  day  bought  a  certain  number  of  sheep 
for  $  100.     By  buying  10  additional  sheep  the  next  day  at 
$  1  less  each,  his  bill  for  all  was  increased  to  $  140.     How 
many  did  he  buy  at  first  ? 

65.  A  party  of  8  hired  a  coach.      If  there  had  been  4 
more  the  expense  would  have  been  reduced  $1  for  each 
person.     How  much  'was  paid  for  the  coach  ? 

66.  A  and  B  agree  to  do  a  piece  of  work  for  $  15.    A  can 
do  the  work  alone  in  8  days ;  B  can  do  it  alone  in  12  days. 
What  should  each  receive  ? 

67.  C  and  D  do  a  piece  of  work  for  $  59.     C  can  do  the 
whole  work  in  2J  weeks,  and  D  can  do  it  in  2%  weeks. 
How  should  the  money  be  divided  between  them? 

68.  A  and  B  can  do  a  piece  of  work  in  10  days.     A  can 
do  it  alone  in  15  days.     They  work  together  4  days,  after 
which  B  finishes  the  work.     If  they  earn  $30,  how  much 
should  each  receive  ? 

69.  A  farmer  being  asked  how  many  apple  trees  he  had, 
replied,  "If  I  had  3  times  as  many  and  5  trees  more,  I 
should  have  1358."    How  many  had  he? 

70.  A  man  bought  a  number  of  pigs  for  $  36.     Nine  of 
them  having  died,  he  sold  •§•  of  the  remainder  for  cost,  and 
received  $  15.     How  many  did  he  buy  ? 


ORAL   EXERCISES.  349 

71.  A  fox  is  60  leaps  ahead  of  a  hound,  and  takes  4  leaps 
while  the  hound  takes  3 ;  but  1  of  the  hound's  equals  2  of 
the  fox's  leaps.     How  many  leaps  must  the  hound  take  to 
catch  the  fox  ? 

72.  A  fox  has  120  rods  the  start  of  a  hound.     If  the 
hound  runs  30  rods  while  the  fox  runs  26,  how  far  will  the 
hound  run  before  he  overtakes  the  fox  ? 

73.  A  hare  is  70  leaps  before  a  hound,  and  takes  5  leaps 
while  the  hound  takes  3;  but  three  of  the  hound's  leaps 
equal  7  of  the  hare's.      How  many  leaps  will  the  hound 
take  to  catch  the  hare  ? 

74.  A  teacher  took  some  pupils  on  an  excursion,  and 
after  expending  15  cents  for  each  pupil,  found  that  she 
had   $2   left.      If   she   had  expended  20  cents  for  each, 
she  would  have  had  only  $1   left.      How   many  pupils 
were  there? 

75.  A  man  bought  a  horse,  a  cow,  and  a  sheep  for  a 
certain  sum.      The  horse  and  the  sheep  cost  5  times  as 
much  as  the  cow,  and  the  sheep  and  the  cow  cost  f-  as  much 
as  the  horse.     What  did  each  cost,  if  the  cow  cost  $  30  ? 

76.  A  lady  had  money  enough  to  pay  15  cents  a  yard 
for  some  ribbon  and  have  50  cents  left.     If  she  had  paid 
25  cents  a  yard  for  it,  she  would  have  needed  50  cents 
more.      How  many  yards  were  there,  and  how  much  money 
had  she  ? 

77.  The  head  of  a  fish  is  8  inches  long.     The  tail  is  as 
long  as  the  head  and  i  of  the  body,  and  the  body  is  as  long 
as  the  head  and  tail.     What  is  the  length  of  the  fish  ? 

78.  A  tree  is  broken  into  three  pieces.     The  part  stand- 
ing is  8  ft.  long.      The  top  piece  is  as  long  as  the  part 
standing  and  £  of  the  middle  piece,  and  the  middle  piece  is 
twice  as  long  as  the  other  pieces.     How  high  was  the  tree  ? 


350  GENERAL  REVIEW   EXERCISES. 

WRITTEN   EXERCISES. 

469.   1.    The  minuend  is  21,870,  and  the  remainder  6492. 
What  is  the  subtrahend  ? 

2.  The  quotient  is  3217,  the  divisor  63,  and  the  re- 
mainder 29.     What  is  the  dividend  ? 

3.  Multiply  7.64  by  .000302, 

4.  Divide  .0085604  by  2.07. 

5.  Simplify  (if  +  4  of  1L  _  ±y  2ft. 

6.  If  |  of  a  yard  of  broadcloth  costs  $3f,  what  will  5| 
yards  cost  ? 

7.  A  farmer  sold  7  firkins  of  butter,  each  containing 
100  pounds,  for  24  cents  per  pound,  and  16  dozen  eggs  for 
18  cents  a  dozen.     He  received  in  payment  50  pounds  of 
sugar  at  5J  cents  per  pound,  18  yards  of  cloth  at  $  1.37-J-  per 
yard,  and  the  rest  in  money.     How  much  money  did  he 
receive  ? 

8.  Half  of  A's  money  is  equal  to  f  of  B's,  and  A  has 
$  18  more  than  B.     How  much  has  each  ? 

9.  How  many  yards  of  silk,  f  of  a  yard  wide,  will  it  take 
to  line  4J-  yards  of  broadcloth  If  yards  wide  ? 

10.  How  much  must  be  paid  for  3580  Ib.  of  coal,  at 
$  6.50  a  ton  ? 

11.  A  square  lot,  measuring  on  each  side  36.5  yards,  is 
inclosed  by  four  lines  of  galvanized  iron  wire.     Eight  yards 
of  this  wire  weigh  a  pound,  and  the  wire  cost  6  cents  a 
pound.     What  did  the  wire  fence  cost  ? 

12.  A  man  bought  ^  of  a  section  (a  square  mile)  of  land 
for  $2500.     He  sold  f  of  it  at  $  14.50  an  acre,  and  the  rest 
at  $  15.75  an  acre.     How  much  did  he  gain  ? 

13.  Divide  $7.75  among  5  boys  and  4  girls,  and  give 
each  boy  J  as  much  as  each  girl. 


WRITTEN  EXERCISES.  351 

14.  If  a  man  takes  2  steps  of  30  inches  each,  in  3  seconds, 
how  long  will  it  take  him  to  walk  10  miles  ? 

15.  Three  men  engage  to  reap  a  field  of  wheat.     A  can 
do  it  in  12  days,  B  in  15  days,  and  C  in  18  days.     In  what 
time  can  they  do  it  together  ? 

16.  What  will  it  cost  to  lay  a  pavement  40  feet  long  and 
9  feet  6  inches  wide,  at  35  cents  a  square  yard  ? 

17.  A  boy  bought  a  certain  number  of  oranges  at  the  rate 
of  5  for  6  cents,  and  sold  them  at  the  rate  of  3  for  5  cents. 
He  gained  70  cents.     How  many  did  he  buy  ? 

18.  A  man  bequeathed  ^  of  his  estate  to  his  wife,  J  to  each 
of  three  children,  -^  to  his  brother,  and  the  rest,  amounting 
to  $  1850,  to  a  charitable  institution.     How  much  was  his 
estate  worth  ? 

19.  Mr.  A  bought  480  bushels  of  grain,  consisting  of 
wheat,  oats,  and  corn,  in  the  proportion  of  3,  4,  and  5. 
How  many  bushels  of  each  did  he  buy  ? 

20.  How  many  acres  are  there  in  a  rectangular  piece  of 
land,  8450  feet  long  and  3580  feet  wide  ? 

21.  How  many  cords  of  wood  are  there  in  a  pile  80  feet 
long,  5  feet  high,  and  4  feet  wide  ? 

22.  A  man  invested  -f  of  his  capital  in  bank  stock,  }  of 
the  remainder  in  real  estate,  and  had  $  4260  left.     What 
was  his  capital  ? 

23.  What  must  be  the  length  of  a  plot  of  ground,  if  the 
breadth  is  18|  feet,  that  its  area  may  contain  56  square  yards  ? 

.24.    If  14  ounces  of  wool  make  2J  yards  of  cloth  1  yard 
wide,  how  much  will  it  take  to  make  6^-  yards  1 J  yards  wide  ? 

25.  One  fourth  of  a  certain  number  is  132  more  than  £ 
of  it.     What  is  the  number  ? 

26.  If  to  a  certain  number  you  add  £  of  itself  and  -J-  of 
itself,  the  sum  will  be  943.     What  is  the  number  ? 


352  GENERAL  REVIEW  EXERCISES. 

27.  A  paid  $  65  an  acre  for  his  farm,  which  was  -jj-  as 
much  as  B  paid  per  acre  for  his  farm  of  160  acres.     What 
was  the  cost  of  B's  farm  ? 

28.  A  man  owns  f  of  a  ship,  and  sells  %  of  his  share  for 
£  1260.     At  this  rate,  what  is  the  value  of  the  ship  in  U.  S. 
money  ? 

29.  If  a  train  runs  30  miles  per  hour,  what  is  its  average 
speed  per  second  ? 

30.  A,  B,  and  C  hire  a  pasture  for  $  155.     A  puts  in  20 
oxen  for  5^-  months,  B  8  oxen  and  28  sheep  for  6  months, 
and  C  56  sheep  for  6|-  months.     If  2  oxen  eat  as  much  as  7 
sneep,  how  much  should  each  man  pay  ? 

31.  What  is  the  interest  on  a  note  for  $460  for  3  yr. 
5  mo.  23  da.  at  5%  ? 

32.  A  merchant  sold  a  quantity  of  goods  at  a  gain  of 
20%.     If,  however,  he  had  purchased  the  goods  for  $  60  less 
than  he  did,  his  gain  would  have  been  25%.     What  did  the 
goods  cost  ? 

33.  A  regiment  of  soldiers,  consisting  of  1100  men,  was 
furnished  with  bread  sufficient  to  last  it  8  weeks,  allowing 
each  maa  15  oz.  per  day.     If  -J-  of  it  was  found  to  be  unfit 
for  use,  how  many  ounces  per  day  should  each  man  receive 
so  that  the  balance  may  last  8  weeks  ? 

34.  If  5  oxen  or  7  horses  will  eat  up  the  grass  of  a  field 
in  60  days,  in  what  time  will  3  oxen  and  4  horses  eat  it  ? 

35.  If  I  buy  coal  at  $4  per   ton  on  4  months'  credit, 
at  what  price  must  I  sell  it  immediately  to  gain  20%,  money 
being  worth  6%  ? 

36.  A  dealer  bought  flour  for  $900  cash,  and  sold  it 
for  $  1080  on  6  months'  credit,  for  which  he  received  a  note. 
If  he  should  get  the  note  discounted  at  a  bank  at  6%,  what 
would  be  the  gain  on  the  flour  ?     (Allow  days  of  grace.) 


WRITTEN  EXERCISES.  353 

37.  A  farmer  was  offered  $  1.45  per  bushel  for  his  wheat, 
but  he  determined  to  have  it  ground  and  sell  the  flour.    It  cost 
to  take  it  to  the  mill  2  J  cents  per  bushel ;  the  miller  took  J- 
f or  grinding ;  it  took  4$ -§•  bushels  to  make  a  barrel  of  flour ; 
he  paid  45  cents  apiece  for  barrels,  and  it  cost  25  cents  per 
barrel  commission  to  sell  it.    75  barrels  were  sold  for  $  550 
and  25  barrels  for  $165.     If  the  refuse  was  sold  for  $100, 
did  he  gain  or  lose,  and  how  much  per  hundred  barrels  ? 

38.  How  many  more  feet  of  fencing  will  be  required  to 
inclose  a  rectangular  field  80  rods  long  and  45  rods  wide, 
than  to  inclose  a  square  one  of  the  same  area  ? 

39.  What  is  the  difference  between  the  true  and  the 
bank  discount  of  $  360  due  in  6  months  at  6%  ?    (No  grace.) 

40.  An   agent    sells    for   a  manufacturer   goods   to  the 
amount  of  $1675.80,  at  a  commission  of  5  %,  and  purchases 
for  him  raw  material  amounting  to  $3860,  at  a  commission 
of  2|  %.     What  is  the  sum  earned  by  the  agent? 

41.  A  carriage  has  its  hind  wheel  4J  ft.  high,  and  its  fore 
wheel  4  ft.  high.    While  the  hind  wheel  is  making  720  revo- 
lutions, how  many  does  the  fore  wheel  make  ? 

42.  An  estate  is  divided  among  three  heirs,  A,  B,  and  C, 
so  that  A  has  -f^  of  the  whole,  and  B  has  twice  as  much  as 
C.     It  is  found  that  A  has  56  acres  more  than  C.     How- 
large  is  the  estate  ? 

43.  What  is  the  present  worth  of  a  debt  of  $  1200,  due 
in  2  years  and  3  months,  without  interest,  money   being 
worth  6  %  ? 

44.  From  your  knowledge  of  circular  measure  and  of  the 
length  of  a  degree  of  longitude  at  the  equator,  compute  the 
circumference  of  the  earth  at  the  equator. 

45.  A  and  B  in  partnership  gained  $860,  of  which  A's 
share  was  $500.     B's  stock  was  $1800.     What  was  A's 
stock  ? 

STAND.  AR. 23 


354  GENERAL   REVIEW  EXERCISES. 

46.  How  many  square  inches  are  there  in  the  entire  sur- 
face of  a  cube  whose  edge  is  17  inches  ? 

47.  A  drover   bought  a  certain  number  of  horses   for 
$  2850.     If  he  had  bought  13  more,  at  $  12  more  each,  they 
would  have  cost  $  4956.     How  many  did  he  buy  ? 

48.  A  man  bought  20  bushels  of  wheat  and  15  bushels  of 
corn  for  $  36,  and  15  bushels  of  wheat  and  25  bushels  of  corn 
for  $32.50.     What  did  he  pay  per  bushel  for  each  ? 

49.  What  should  be  the  cost  of  15  crates  of  berries,  each 
containing  2|  pecks,  at  9|  cents  a  quart  ? 

50.  What  will  be  the  expense  of  plastering  a  room  18  feet 
long,  15  feet  wide,  and  8  feet  high,  at  $  .30  a  square  yard, 
allowing  150  square  feet  for  doors,  windows,  etc.  ? 

51.  For  what  sum  must  a  note  be  drawn  at  3  months  to 
net  $  150,  when  discounted  at  6%  ?    (Allow  days  of  grace.) 

52.  At  3^-  cents  a  foot,  board  measure,  what  is  the  cost 
of  5  pieces  of  sawed  timber,  each  measuring  18  feet  long, 
1  foot  4  inches  wide,  and  11  inches  thick  ? 

53.  A  man  wishes  his  son  to  have  $  3000  when  he  is  21 
years  of  age.     What  sum  must  be  deposited  at  the  son's 
birth,  in  a  savings  bank  which  pays  compound  interest  at 
the  annual  rate  of  6%,  so  that  the  deposit  shall  amount  to 
that  sum  when  the  boy  becomes  of  age  ? 

54.  What  is  the  entire  surface  of  a  cube,  the  contents  of 
which  are  15,625  cubic  feet  ? 

55.  How  many  acres  of  land  are  there  in  a  rectangular 
farm  -|  of  a  mile  long  and  ^  of  a  mile  broad  ? 

56.  In  what  time  will  $300,  at  6%  simple  interest,  yield 
an  amount  of  interest  equal  to  the  principal  ? 

57.  How  far  from  the  base  of  a  building  must  a  ladder 
50  feet  long  be  placed  to  reach  a  window  40  feet  from  the 
ground  ? 


WRITTEN   EXERCISES.  355 

58.  What  will  it  cost  to  dig  a  cellar,  38  feet  long,  30  feet 
wide,  and  8  feet  deep,  at  $  .45  a  cubic  yard  ? 

59.  What  must  be  the  price  paid  for  5%  stock  so  that 
it  may  yield  the  same  rate  of  income  as  41%  stock  at  96  ? 

60.  How  many  bushels  of  wheat  will  a  bin  hold  that  is 
5  feet  long,  4-J-  feet  wide,  and  4  feet  deep  ? 

61.  If  $8000  worth  of  41%  stocks  are  sold  at  87£,  and 
the  proceeds  are  invested  in  6%  stock  at  116J-,  what  will  be 
the  change  in  the  income  ? 

62.  A,  B,  and  C  agree  to  build  a  house.     A  and  B  can  do 
the  work  in  32  days,  B  and  C  in  28  days,  and  A  and  C  in 
26  days.      How  long  will  it  take  them  to  do  it  working 
together  ?     How  long  will  it  take  each  to  do  it  alone  ? 

63.  How  many  barrels  of  water  will  a  rectangular  tank 
contain  which  is  7-j-  feet  long,  4  feet  wide,  and  2j-  feet  deep? 

64.  If  8  men  spend  $  32  in  15  weeks,  how  much  will  56 
men,  at  the  same  rate,  spend  in  a  year  ? 

65.  If  12  men  can  build  a  wall  30  feet  long,  6  feet  high, 
and  3  feet  thick,  in  15  days,  by  working  12  hours  per  day, 
in  what  time  will  60  men  build  a  wall  300  feet  long,  8  feet 
high,  and  6  feet  thick,  when  they  work  only  8  hours  a  day  ? 

66.  If  a  pipe  1^-  in.  in  diameter  fills  a  cistern  in  2  hours, 
how  long  will  it  require  a  pipe  that  is  3  in.  in  diameter  to 
fill  it,  no  allowance  being  made  for  the  difference  in  friction  ? 

67.  A  man  sold  two  farms  for  $4800  each.     On  the  one 
he  gained  20%,  and  on  the  other  he  lost  20%.    Did  he  gain 
or  lose  on  the  sale,  and  how  much  ? 

68.  B  owned  75  shares  of  stock  in  a  building  association, 
at  $50  each.     The  association  declared  a  dividend  of  8%, 
payable  in  stock.     How  many  shares  did  he  then  own? 

69.  Mr.  W.  bought  40  shares  of  stock,  $50  each,  at  2|% 
discount.     He  sold  ^  of  it  at  \°/0  discount,  and  the  rest  at 
If  %  premium.     What  was  his  gain  ? 


356  GENERAL  REVIEW  EXERCISES. 

70.  A  man  traveled  3  days  at  the  rate  of  18  miles  per 
day,  4  days  at  the  rate  of  22  miles  per  day,  and  5  days  at 
the  rate  of  28  miles  per  day.     What  was  his  average  rate  of 
travel  per  day  ? 

71.  A  liveryman  borrowed  money  at  6%  to  purchase  a 
horse.     The  horse,  which  cost  him  $90,  earned  $1  a  day, 
and  the  expense  of  keeping  him  each  day  was  \°/0  of  the 
purchase  price.     At  the  end  of  a  year  the  liveryman  sold 

1the  horse  for  $70.     How  much  did  he  gain,  if  the  horse 
worked  312  days  during  the  year  ? 

72.  A  tank  which  holds  200  gallons  can  be  filled  by  one 
pipe  in  15  minutes  and  emptied  by  another  pipe  in  40  min- 
utes.    If  the  tank  is  empty  and  both  are  opened  at  the 
same  time,  how  long  will  it  take  to  fill  it  ? 

73.  In  the  Centigrade  and  Fahrenheit  thermometers  the 
freezing  points  are  0°  and  32°  respectively,  and  the  boiling 
points  100°  and  212°  respectively.     When  the  temperature 
is  50°  Fahrenheit,  what  temperature  will  the  Centigrade 
thermometer  indicate  ? 

74.  An  agent  sold  goods  at  a  commission  of  5%  through 
a  broker  who  charged  him  2%,  and  the  agent's  commission 
after  paying  the  brokerage  was  $  315.     How  much  did  the 
agent  remit  to  his  employer  ? 

75.  How  much  will  be  realized  from  the  sale  of  a  draft 
for  $  1200  sold  at  \%  premium  ? 

76.  What  will  be  the  face  of  a  60-day  draft  purchased 
if  or  $450,  if  the  rate  of  exchange  is  \°/0  premium  and  the 
rate  of  discount  6  °]0  ?     (Allow  days  of  grace.) 

77.  A  room   is  18  feet  long,  15  feet  wide,  and  9  feet 
high.     What  must  be  the  length  of  a  line  extending  from 
one  of  the  lower  corners  to  an  opposite  upper  corner  ? 

78.  A  and  B  together  have  $  851.     How  much  has  each 
if  -fy  of  A's  money  equals  f  of  B's  ? 


WRITTEN  EXERCISES.  357 

79.  A  detachment  of  2000  soldiers  was  supplied  with 
bread  sufficient  to  last  12  weeks,  allowing  each  man  14 
ounces   a  day ;    but   105   barrels,    containing   200   pounds 
each,  were  wholly  spoiled.    How  much  a  day  may  each  man 
eat,  that  the  remainder  may  supply  them  12  weeks  ? 

80.  Find  the  value  of  (3.0005  x  .006) -f-  .0009. 

81.  An   agent  sold  a  house   at  2%   commission.      He 
invested  the  net  proceeds  of  the  sale  in  city  lots,  after  de- 
ducting his  commission  of  3%  for  buying  them,  and  found 
that  his  commissions  amounted  to  $350.     For  how  much 
was  the  house  sold? 

82.  What  is  the  area  in  acres  of  a  rectangular  field  whose 
breadth  is  65  ch.  20  1.,  and  whose  length  is  70  ch.  18  1.  ? 

83.  A  farmer  mixed  10  bushels  of  oats,  worth  $.35  a 
bushel,  12  bushels  of  corn,  worth  $.60  a  bushel,  and  7 
bushels  of  rye,  worth  $  .78  a  bushel.    What  was  the  average 
value  of  a  bushel  of  the  mixture  ? 

84.  In  a  mixture  of  gold  and  silver,  weighing  64  ounces, 
there  are  4  ounces   of  silver.     How  much  gold  must  be 
added  that  there  may  be  f  of  an  ounce  of  silver  in  18  ounces 
of  the  mixture  ? 

85.  There  is  a  circular  park  250  rods  in  diameter,  and 
within  it  is   a   circular   lake   125   rods   in   circumference. 
What  is  the  area  of  the  park  exclusive  of  the  lake  ? 

86.  How  many  balls,  3  in.  in  diam.,  will  weigh  as  much 
as  a  ball  of  the  same  material  and  density  9  in.  in  diam.  ? 

87.  A  note,  due  in  4  months,  dated  March  4,  was  dis- 
counted  May   15.      For   what   time   was    it   discounted? 
(Allow  days  of  grace.) 

88.  A  man  bought  a  house  for  $2700.     He  repaired  it 
for  a  tenant  who  agreed  to  pay  him  a  yearly  rent  of  $  225, 
which  was  15%  less  than  the  cost  of  repairing  it.     What 
was  the  entire  cost  of  the  house  ? 


358  GENERAL  REVIEW  EXERCISES. 

89.  Three  men  took  a  contract  to  build  a  bridge  for 
$32,525.     The  first  had  80  men  at  work  for  40  days,  the 
second  had  70  men  at  work  for  45  days,  and  the  third  had 
56  men  at  work  for  50  days.     The  third  received  $  500  for 
superintending  the  work.     How  much  was  each  one's  share 
of  the  contract  price  ? 

90.  It  cost  $150  to  support  4  grown  persons  and  3  chil- 
dren 8  weeks.    What  will  it  cost  to  support  3  grown  persons 
and  8  children  for  the  same  time,  if  3  children  cost  as  much 
as  2  grown  persons  ? 

91.  A  man  has  real  estate  assessed  at  $  3200,  and  personal 
property  at  $1280.     If  he  pays  a  tax  of  If  %,  what  is  his 
total  tax? 

92.  If   a  tax   of   $60   is   paid   on  a  factory  valued  at 
$12,000,  what  is   the   assessed  valuation   of   a   residence 
that  is  taxed  $8.87-|  at  the  same  rate? 

93.  What  is  the  ad  valorem  duty,  at  40%,  upon  a  con- 
signment of  425  dozen  silk  handkerchiefs,  invoiced  at  35 
francs  per  dozen  ? 

94.  A  merchant  imported  40  pieces  of  carpet,  each  piece 
containing  56  square  yards,  invoiced  at  3s.  Sd.  per  square 
yard,  upon  which  he  paid  a  specific  duty  of  15  cents  per 
square  yard,  and  30%   ad  valorem.     What  was  the  total 
amount  of  duty  paid  ? 

95.  Two  men  were  employed  to  build  a  wall.     The  first 
received  $.87£  per  day,  and  the  second  $1.12£  per  day. 
Each  worked  until  he  had  earned  $60.     How  many  days 
did  each  labor  ? 

96.  If  $300,  placed  at  interest,  yields  an  income  of  $18 
in  9  months,  how  much  must  be  placed  at  interest,  at  the 
same  rate,  to  yield  an  income  of  $  115  in  6  months  ? 

97.  A,  B,  and  C  entered  into  partnership.     A  advanced 
$1200.  B,  $900,  and  C,  $850.     A  left  his  money  in  the 


WRITTEN  EXERCISES.  359 

business  8  months,  B,  10  months,  and  C,  12  months.     They 
gained  $  1296.    To  what  share  of  the  profit  is  each  entitled  ? 

98.  A  and  B  have  the  same  income.     A  saves  £  of  his, 
but  B,  by  spending  $  100  each  year  more  than  A,  at  the 
end  of  5  years  finds  himself  $  240  in  debt.     What  was  the 
income  of  each  ? 

99.  A  ladder  70  feet  long  is  so  planted  as  to  reach  a 
window  40  feet  from  the  ground,  on  one  side  of  the  street, 
and  without  moving  it  at  the  foot,  it  will  reach  a  window  30 
feet  high  on  the  other  side.     What  is  the  width  of  the 
street  ? 

100.  A  father  gave  to  his  four  sons  $  1200,  which  they 
were  to  divide  so  that  each  son  should  receive  $60  more 
than  his  next  younger  brother.     What  was  the  share  of  the 
oldest  ? 

101.  A  and  B  are  on  opposite  sides  of  a  circular  pond 
which  is  1380  feet  in  circumference.     They  walk  around  it, 
starting  at  the  same  time  and  in  the  same  direction.     A 
goes  at  the  rate  of  45  yards  per  minute,  and  B  at  the  rate 
of  50  yards  per  minute.     In  what  time  will  B  overtake  A, 
and  how  many  times  around  the  pond  will  he  have  traveled  ? 

102.  A  father  wishes  to  divide  $1500  between  his  son 
and  daughter,  whose  ages  are  12  and  16  years,  respectively, 
in  such  proportions  that  the  share  of  each,  being  put  at 
simple  interest  at  6%,  will  amount  to  the  same  sum  when 
they  reach  the  age  of  21.     What  should  each  receive  ? 

103.  A,  B,  and  C  are  to  share  $  1200  in  the  proportion 
of  3,  4,  and  5,  respectively.     B  dies.     How  should  the  whole 
sum  be  divided  between  A  and  C  ? 

104.  A  man  bought  a  house,  a  store,  and  a  lot.     The  lot 
cost  $  1650,  the  house  and  store  5f  times  as  much  as  the 
lot,  and  the  store  cost  \  as  much  as  the  house  and  lot. 
What  was  the  cost  of  each  ? 


360  GENERAL  REVIEW  EXERCISES. 

105.  At  $  75  an  acre,  and  $1.50  a  rod  for  fencing,  what 
will  it  cost  me  to  purchase  and  fence  a  field  having  two 
parallel  sides  100  and  80  rods  long,  respectively,  the  distance 
between  them  being  70  rods,  and  the  other  two  sides  being 
equal  ? 

106.  A  person  in  purchasing   sugar   found  that   if  he 
bought  sugar  at  5  cents,  he  would  lack  30  cents  of  having 
money  enough  to  pay  for  it,  so  he  bought  sugar  at  4i  cents, 
and  had  45  cents  left.     How  many  pounds  did  he  buy  ? 

107.  What  is  the  base  of  a  triangular  field  whose  area  is 
1  acre  65  sq.  rods,  and  whose  altitude  is  18  rods  ? 

108.  What  is  the  altitude  of  a  triangular  plot  of  ground 
whose  area  is  5|-  acres,  and  whose  base  is  44  rods  ? 

109.  A  man  sold  a  horse  and  carriage  for  $  597,  gaining 
by  the  sale  25%  on  the  cost  of  the  horse  and  10%  on  the 
cost  of  the  carriage.     If  f  of  the  cost  of  the  horse  equaled  -| 
of  the  cost  of  the  carriage,  what  was  the  cost  of  each  ? 

110.  A  merchant  bought  cloth  to  the  amount  of  $750, 
and   silk   goods   to   the   amount   of   $500.     On  the  cloth 
goods  he  gained  20%  and  on  the  silk  he  lost  16f  %.     How 
much  did  he  gain  ? 

111.  If  a  house  is  bought  for  $2150,  and  sold  again  for 
$  2365,  what  is  the  gain  per  cent  ? 

112.  A  merchant   sold  a  coat   for   $15.40,  and   gained 
20%.     How  much  would  he  have  gained  had  he  sold  it  for 
$16.50? 

113.  A  owes  B  $1200,  payable  in  6  months,  but  at  the 
end  of  4  months  he  pays  $  400.     How  long  after  this  pay- 
ment is  made  will  it  be  before  the  rest  is  equitably  due  ? 

114.  A  house  is  38  feet  from  the  ground  to  the  eaves. 
How  long  must  a  ladder  be  to  reach  the  eaves  if  its  foot  is 
placed  25  feet  from,  the  house  ? 


WRITTEN  EXERCISES.  361 

115.  A  farmer  had  his  sheep  in  three  fields,     -f  of  the 
number  in  the  first  field  was  equal  to  f  of  the  number  in 
the  second  field,  and  -J  of  the  number  in  the  second  field 
was  f  of  the  number  in  the  third  field.     If  the  entire  num- 
ber was  434,  how  many  were  there  in  each  field  ? 

116.  If  9  men  can  mow  75  acres  of  grass  in  6  days  of  8J- 
hours  each,  in  how  many  days  of  8  hours  each  can  15  men. 
mow  198  acres  ? 

117.  A  board  is  18  feet  long,  20  inches  wide  at  one  end, 
and  tapers  gradually  until  it  is  only  1  foot  wide  at  the 
other  end.     It  is  1  inch  thick.     How  many  board  feet  does  it 
contain  ? 

118.  What  is  the  compound  interest  of  $  1650  for  3  years 
at  6%,  interest  being  compounded  semi-annually  ? 

119.  In  what  time  will  $460.75  earn  $95  interest  when 
the  rate  is  6%  ? 

120.  What  is  the  rate  per  cent  per  annum  when  $  712  in 
3  years  4  months  earns  $142.40  ? 

121.  A,  B,  and  C  in  partnership  gained  $3192.      A's 
stock  was  $  5600,  which  was  1^  times  B's,  and  B's  was  1£ 
times  C's.     What  was  the  gain  of  each  ? 

122.  D,  E,  and  F  engage  to  do  a  piece  of  work  for  $  381. 
D  sends  7  men  6  days,  E  sends  8  men  5  days,  and  F  sends 
5  men  9  days.     What  should,  each  receive  ? 

123.  A's  capital  was  in  trade  9  months,  B's  12  months, 
and  C's  15  months.     A's  gain  was  $  1125,  B's  $  1200,  and 
C's  $1275.     The  whole  capital  was  $18,600.     What  was  the 
capital  of  each  ? 

124.  A  man  bought  a  farm  for  $2500  and  sold  it  for 
$3000.      If  the  buying  and  selling  were  done  by  a  real 
estate  agent  who  charged  2%   for  each  transaction,  what 
per  cent  of  the  purchase  price  did  the*  man  gain? 


862  GENERAL  REVIEW   EXERCISES. 

125.  The  diameter  of  a  circular  plot  for  flowers  is  8  feet. 
"What  must  be  the  diameter  of  a  similar  plot  which  shall 
contain  6-J-  times  as  much  area  ? 

126.  The  weight  of  oak  ashes  is  yf^  of  the  weight  of  the 
wood  consumed,  and  the  weight  of  carbonate  of  potash  con- 
tained in  the  ashes  is  .065  of  the  weight  of  the  ashes.    How 
many  ounces  of  carbonate  of  potash  are  there  in  the  ashes 
of  500  pounds  of  oak  wood  ? 

127.  What  is  the  G.  C.  D.  of  3038,  5394,  and  8308? 

128.  What  is  the  L.  C.  M.  of  42,  63,  49,  91,  and  70  ? 

129.  A,  B,  and  C  together  have  $4750.     How  much  has 
each  if  A  has  f  as  much  as  B,  and  C  fy  as  much  as  both  A 
andB? 

130.  A  speculator  bought  stock  at  25%  below  par  and 
sold  it  at  20%  above  par.     He  gained  $1560.     How  much 
did  he  invest  ? 

131.  A  carriage  maker  sold  two  carriages  for  $300  each. 
On  one  he  gained  25%  ;  on  the  other  he  lost  25%.     Did  he 
gain  or  lose  by  the  sale  ?     How  much,  and  how  much  per 
cent  ? 

132.  New  York  is  74°  3'  west  of  Greenwich,  England. 
What  is  the  difference  in  time  between  the  two  places  ? 

133.  Paris  is  about  2°  20'  east  of  Greenwich.     What  is 
the  difference  in  time  between  'New  York  and  Paris  ? 

134.  Which  is  the  heavier,  and  how  much,  an  ounce  of 
lead  or  an  ounce  of  gold  ? 

135.  A  barn  worth  $900  was  insured  for  f  of  its  value 
for  $  3.75.     What  was  the  rate  of  insurance  ? 

136.  A  ship  worth  $48,000,  and  its  cargo  valued  at  \ 
that  amount,  were  insured  for  -f  of  their  value,  at 
What  was  the  cost  "of  the  insurance  ? 


WRITTEN"   EXERCISES.  363 

137.  If  a  ladder,  placed  8  feet  from  the  base  of  a  building 
40  feet  high,  just  reached  the  top,  how  far  must  it  be  placed 
from  the  base  of  the  building  that  it  may  reach  a  point  10 
feet  from  the  top  ? 

138.  The  total  net  weight  of  40  loads  of  hay  was  56,724 
pounds.     What  was  the  hay  worth  at  $>  8.25  per  ton  ? 

139.  A  load  of  four-foot  wood  is  3J  feet  high  and  1\  feet 
long.     What  must  be  paid  for  3  such  loads,  when  wood 
sells  at  $3. 75  per  cord? 

140.  At  the  rate  of  $  12,760  a  mile,  what  will  it  cost  to 
construct  a  railroad  5  miles,  28  rods,  2  yards  long  ? 

141.  If  1  bushel  3  pecks  of  wheat  are  sown  to  the  acre, 
how  much  land  can  be  sown  with  the  contents  of  a  bin  4 
feet  long,  3  feet  wide,  and  2-j-  feet  deep,  filled  with  wheat  ? 

142.  John  and  Charles  can  do  a  piece  of  work  in  45  days, 
and  Charles  can  do  f  as  much  as  John.     In  what  time  can 
each  do  the  work  alone  ? 

143.  A,  B,  and  C  pasture  an  equal  number  of  cattle  upon 
a  field  of  which  A  and  B  are  the  owners,  A  of  9  acres  and 
B  of  15  acres.     If  C  pays  $  24  for  his  pasturage,  how  much 
should  A  and  B  each  receive  ? 

144.  A  and  B  engaged  to  do  a  piece  of  work  for  $  385.     A 
worked  \  as  many  days  as  B,  plus  5  days,  and  received  $  175. 
How  many  days  did  each  work  ? 

145.  A  note  for  $100  was  due  on  Sept.  1st,  but  on  Aug. 
llth  the  maker  proposed  to  pay  as  much  in  advance  as  would 
allow  him  2  months  after  Sept.  1st  to  pay  the  balance.     How 
much  must  be  paid  Aug.  llth,  money  being  worth  6%  ? 

146.  A  and  B  in  partnership  gained  $1200.     A  owned 
•g-  of  the  stock,  plus  f  500,  and  gained  $  600.     What  was  the 
entire  stock  ? 


364  GENERAL    REVIEW    EXERCISES. 

147.  $  360.  MOBILE,  ALA.,  June  24,  1892. 

Three  months  after  date,  for  value  received,  I  promise  to 
pay  D.  C.  Morgan,  or  order,  Three  Hundred  Sixty  Dollars, 
with  interest  at  6%.  A>  B,  HENRY. 

This  note  was  discounted  at  a  bank  at  6%,  July  15, 1892. 
What  were  the  proceeds  ? 

148.  A  hollow  sphere  whose  diameter  is  6  inches  weighs 
-j-  as  much  as  a  solid  sphere  of  the   same  material  and 
diameter.     How  thick  is  the  shell? 

149.  A  train  started  on  a  trip  of  245  miles  at  35  miles  an 
hour,  but  after  having  gone  140  miles  it  was  delayed  20 
minutes.     If  it  finished  the  trip  at  the  rate  of  30  miles  an 
hour,  how  much  behind  time  was  it  ? 

150.  Twenty  per  cent  of  a  barrel  of  oil  leaked  out.    What 
per  cent  must  be  gained  on  the  remainder  that  a  gain  of 
10%  may  be  realized  on  the  cost  of  the  oil  ? 

151.  How  much  alloy  must  be  mixed  with  2  Ib.  2  oz. 
15  pwt.  19  gr.  of  pure  gold,  to  make  gold  18  carats  fine  ? 

152.  A  and  B  traded  with  equal  sums  of  money.      A 
gained  a  sum  equal  to  -i-  of  his  capital,  and  B  lost  $  220.     B 
then  had  %  as  much  as  A.     How  much  capital  had  each  at 
first? 

153.  A  man  invested  |-  of  his  money  in  a  foundry,  ex- 
pended |-  of  what  he  had  left  in  building  a  house,  and  still 
had  $  4671.     How  much  money  had  he  at  first  ? 

154.  If  in  selling  cloth  -|  of  the  gain  is  equal  to  -fa  of  the 
selling  price,  for  how  much  will  3^  yards  sell  that  cost  $5 
per  yard  ? 

155.  A  teacher  agreed  to  teach  9  months  for  $562-1-  and 
his   board.     At  the  end  of  the  term,  on  account  of  two 
months'  absence  caused  by  sickness,  he  received  only  $  409^. 
What  was  his  board  worth  per  month  ? 


WRITTEN  EXERCISES.  365 

156.  A  person  after  spending  $40  more  than  .6  of  his 
money  had  $  60  less  than  .42-?-  of  it  left.     How  much  money 
had  he  at  first  ? 

157.  What  is  the  greatest  number  which  will  divide  27, 
48,  90,  and  174,  and  leave  the  same  remainder  in  each  case  ? 

158.  The  gross  earnings  of  a  mill  were  $  365,816.92.    The 
entire  expenses  exclusive  of  repairs  were  $318,214.84;  the 
repairs  were  .06  of  the  earnings.     If  the  net  profits  were 
divided  equally  among  8  shareholders,  what  was  the  share 
of  each  ? 

159.  A  grain  dealer  expended  a  certain  sum  of  money  in 
the  purchase  of  wheat,  1^-  times  as  much  in  the  purchase  of 
barley,  and  twice  as  much  for  oats.     He  sold  the  wheat  at  a 
profit  of  .05  of  the  cost,  the  barley  at  a  profit  of  .08,  the  oats 
at  a  profit  of  .1,  receiving  for  all  the  grain  $9740.     What 
did  he  pay  for  each  kind  of  grain  ? 

160.  I  wish  to  raise  $  550  by  having  my  note  discounted 
at  a  bank  for  2  mo.  15  da.  at  6%.     What  must  be  the  face 
of  the  note  ?     (Allow  days  of  grace.) 

161.  A  farmer  sold  a  team  of  horses  for  $440,  but  did 
not  receive  his  pay  for  them  until  1  yr.  8  mo.  after  the  sale. 
He  had  at  the  same  time  a  cash  offer  of  $410  for  them. 
Did  he  gain  or  lose  by  the  sale  and  how  much,  money  being 
worth  6%? 

162.  A  weight  of  240  pounds,  suspended  on  a  pole  4  feet 
in  length,  the  point  of  suspension  being  6  inches  from  the 
middle,  is  carried  by  two  men,  the  ends  of  the  pole  resting 
on  their  shoulders.     How  much  of  the  weight  is  borne  by 
each  man  ? 

163.  A  and  B  invested  equal  sums  in  business.     A  gained 
a  sum  equal  to  25%  of  his  stock,  and  B  lost  $225.     A's 
money  at  this  time  was  double  that  of  B's.     What  amount 
did  each  invest  ? 


366  GENERAL  REVIEW  EXERCISES. 

164.  A  owned  two  farms  for  the  better  of  which  he  asked 
50%  more  than  for  the  other.     Not  finding  a  purchaser,  he 
reduced  the  price  of  the  better  33-^%,  and  the  price  of  the 
other  20%,  and  sold  them  both  for  $5580.     What  was  the 
price  asked  for  each  ? 

165.  A  merchant  bought  a  bill  of  goods  amounting  to 
$  3257  on  a  credit  of  3  months,  but  was  offered  a  discount 
of  2-^-%  for  cash.     How  much  would  he  have  gained  by  pay- 
ing cash,  money  being  worth  7%? 

166.  Which  is  better  for  me,  to  buy  6%  bonds  at  72%, 
or  to  invest  my  money  in  mortgages  bearing  8% ?     How 
much  better  is  it  ? 

167.  A  machine  shop  was  insured  at  an  annual  rate  of 
3%,  the  premium  paid  being  $750.     For  how  much  was  it 
insured  ? 

168.  A  merchant  bought  broadcloth  at  a  discount  of  25% 
from  the  marked  price,  receiving  besides  a  discount  of  5% 
for   cash.     If  he   sold  it   at   an   advance  of   10%  on  the 
marked  price,  what  was  his  gain  per  cent  ? 

169.  A  commission  merchant  received  35,000  bushels  of 
oats,  which  he  sold  at  32  cents  per  bushel.     He  was  in- 
structed to  invest  the  proceeds,  together  with  $  4000  cash 
sent  him,  in  prints  at  5^-  cents  per  yard.     If  his  commission 
both  for  buying  and  for  selling  was  2%,  how  many  yards 
of  prints  did  he  buy  ? 

170.  I  find  that  I  owe  A  50%  more  than  I  owe  C,  and 
B  33|%  more  than  I  owe  A.     Now  if  I  owe  B  $800  more 
than  I  do  C,  how  much  is  my  indebtedness  to  each  ? 

171.  A  started  in  business  with  a  capital  of  $4000,  and 
at  the  end  of  5  years  he  was  joined  by  B  with  a  capital 
of  $5000.     Three  years  later  they  were  joined  by  C  with 
a  capital  of  $6000.     If  at  the  end  of  15  years  after  the 
commencement  of  business  the  profits,  which  amounted  to 
$  18,240,  were  divided,  how  much  was  each  one's  share  ? 


WRITTEN   EXERCISES.  367 


172.  How  many  shares  of  stock,  at  113£,  can  a  broker 
purchase  for  me  with  $22,675,  brokerage  \%  ? 

173.  I  am  offered  6%  stock  at  84,  and  5%  stock  at  72. 
Which  investment  is  preferable,  and  how  much  ? 

174.  I  am  desirous  of  securing  an  income  of  6^%  or  1°/0 
on  my  investments.     Can  I  do  it  by  purchasing  5%  stock 
at  75%  ?     What  will  be  the  rate  of  income  ? 

175.  I  have,  as  the   net  proceeds  of  a  consignment  of 
goods  sold  by  me,  $3816.48,  which  the  consignor  desires 
me  to  remit  by  draft  at  2  months.    If  the  rates  of  exchange 
are  f  %  premium,  and  the  rate  of  interest  6%,  what  will  be 
the  face  of  the  draft  ?     (Allow  days  of  grace.) 

176.  A  man  bought  a  horse  and  a  carriage,  paying  twice  as 
much  for  the  horse  as  for  the  carriage.     He  sold  them  both 
for  $  662,  receiving  15%  more  for  the  hprse,  and  8%  more  for 
the  carriage  than  they  cost  him.     What  did  they  each  cost 
him? 

177.  A  man  sold  500  acres  of  land,  receiving  in  payment 
|  of  the  value  in  cash,  and  the  rest  in  a  note  due  in  3 
months  without  interest.     He  immediately  discounted  the 
note  at  a  bank  at  6%,  paying  $57.50  discount.     What  was 
the  price  of  the  land  per  acre  ?    (Allow  days  of  grace.) 

178.  How  many  slates  will  be  required  to  cover  a  roof, 
each  side  of  which  is  34  feet  9  inches  long  and  16  feet  wide, 
allowing  4  slates  to  cover  a  square  foot  ;  and  what  will  they 
cost  at  the  rate  of  $  4.75  per  C  ? 

179.  An  article  was  sold  at  a  price  which  was  |  above 
cost.     If  the  cost  had  been  £  of  what  it  really  was  and  the 
selling  price  had  remained  the  same,  the  gain  would  have 
been  $6.75.     How  much  did  the  article  cost? 

180.  Three  men  bought  a  grindstone  20  inches  in  diameter. 
How  much  of  the  diameter  must  each  grind  off  so  as  to  share 
the  stone  equally,  making  no  allowance  for  the  eye  ? 


AVERAGE   OF   PAYMENTS. 


470.  1.   How  long  may  $  1  be  kept  to  balance  the  use  of 
$  5  for  2  months  ?     $  6  for  3  months  ?     $10  for  4  months  ? 

2.  How  long  may  $  10  be  kept  to  balance  the  use  of  $  6 
for  5  months  ?     $  8  f or  10  months  ?     $  12  for  5  months  ? 

3.  I  owe  B  two  equal  debts,  one  due  in  3  months  and  the 
other  in  6  months.    When  may  I  pay  both  at  one  payment  ? 

4.  If  I  pay  $20  3  months  before  it  is  due,  how  long 
after  it  is  due  may  I  keep  $  30  to  balance  it  ? 

5.  If  I  owe  $20  due  in  4  months  and  $40  due  in  6 
months,  at  what  time  are  both  debts  equitably  due  ? 

471.  Finding  the  equitable  time  for  discharging,  by  one 
payment,  sums  due  at  different  times  is  Averaging  Payments. 

472.  The  date  at  which  the  debts  may  be  equitably  dis- 
charged by  a  single  payment  is  the  Average  Time. 

473.  The  time  that  must  elapse  before  the  debt  becomes 
due  is  the  Term  of  Credit. 

474.  The  time  that  must  elapse  before  the  debts  due  at 
different   times   may  be  equitably  discharged  by  a  single 
payment  is  the  Average  Term  of  Credit. 

475.  When  the  terms  of  credit  begin  at  the  same  date. 

1.  A.  T.  Stewart  &  Co.  sold  a  bill  of  goods  upon  the  fol- 
lowing terms  :  $  400  cash,  $  300  due  in  2  months,  and  $  400 
due  in  4  months.  At  what  time  might  the  whole  indebted- 
ness be  equitably  discharged  by  a  cash  payment  ? 


DEFINITIONS.  369 

EXPLANATION.  — 

$  400  for  0  mo.  =  $  1  for  -  .  Since  $400  was  to  be 

300  for  2  mo.  =  $  1  for    600  mo.  paid  in  cash,  there  was 


400  for  4  mo.  =  $  1  for  1600  mo.          f.°  *erm 
_  _  .  that  sum.  Since  $300 

$  1100  2200  mo.          was  to  be  Paid  in  2 

months,  the    use  of 

2200  mo.  -f-  1100  =  2  m.0.    Average  term  of  credit,   that  Sum  for  2  months 

is  equal  to  the  use  of 

$  1  for  600  months  ;  and  the  use  of  $  400  for  4  months  is  equal  to  the 
use  of  $  1  for  1600  months.  Hence,  the  credit  of  the  whole  debt,  $  1100, 
is  equal  to  the  credit  of  $  1  for  2200  months,  or  $  HOC  for  y^  part  of 
2200  months,  which  is  2  months,  the  average  term  of  credit. 

RULE.  —  Multiply  each  debt  by  its  term  of  credit,  and  divide 
the  sum  of  the  products  by  the  sum  of  the  debts.'  The  quotient 
will  be  the  average  term  of  credit. 

Disregard  fractional  parts  of  a  day  that  are  less  than  £,  and  con- 
sider ^  of  a  day  or  more  as  a  whole  day. 

2.  Find  the  average  term  of  credit  of  a  bill  of  goods 
amounting  to  $2300,  on  the  following  terms:  $300  cash, 
$  1200  due  in  3  months,  and  the  balance  due  in  4  months. 

3.  Marshall  Field  sold  a  bill  of  goods  payable  as  follows  : 
$  500  in  1  month,  $  500  in  2  months,  and  $  800  in  4  months. 
What  was  the  average  term  of  credit  ? 

4.  Whitney  &  Co.  sold  a  bill  of  lumber  on  the  following 
terms:  $1500  cash,  $3000  payable  in  30  days,  and  $2000 
payable  in  90  days     What  was  the  average  term  of  credit  ? 

5.  Thurber,  Whyland  &  Co.  sold  to  F.  N.  Burt  a  bill  of 
goods  amounting  to  $2400,  payable  as  follows:  %  in  30 
days,  £  the  remainder  in  60  days,  and  the  balance  in  4 
months.     What  was  the  average  term  of  credit? 

6.  Mr.  Birge  bought  a  bill  of  goods  amounting  to  $  3000, 
payable  as  follows  :  \  in  3  months,  J  in  2  months,  and  the 
rest  in  4  months.     What  was  the  average  term  of  credit  ? 

STAND.    AR.  —  24 


370  AVERAGE  OF  PAYMENTS. 

476.  When  the  terms  of  credit  begin  at  different  dates. 

1.  Find  the  average  time  of  payment  of  the  following 
bills  :  February  10,  1891,  $  400  due  in  2  months ;  March  15, 
1891,  $350  due  in  3  months ;  and  April  12,  1891,  $300  due 
in  3  months. 

EXPLANATION. 

$400  due  April  10.     400  -Adding  to  the 

350  due  June  15.     350  x  66  =  23100       date  of  the  pur- 

300  due  July    12.    300  x  93  =  2790Q       chase  of  each  bil1 
its  term  of  credit, 

1050  51000        we    obtain    the 

time  when  it  is 

51000  da.  -*- 1050  =  48if  days.  due,  and  so  we 

April  10  +  49  days  =  May  29,  average  term,    have   $400    due 

April    10,   $350 

due  June  15,  $  300  due  July  12.  The  average  time  when  the  bills  will 
be  due  will  be  either  after  the  earliest  date,  or  before  the  latest  date, 
and  so  we  may  select  either  of  these  dates  from  which  to  compute  the 
average  time.  Selecting  the  earliest  date,  we  find  that  $350  was  due 
•66  days  after  that  time,  and  $  300  was  due  93  days  thereafter.  Averag- 
ing, as  in  Art.  475,  we  find  the  term  of  credit  to  be  48£f  days,  and 
since  the  fraction  ^  is  greater  than  ^,  the  term  of  credit  is  considered 
to  be  49  days.  This,  added  to  April  10,  gives  May  29,  the  average 
time  of  payment. 

RULE.  —  Select  the  earliest  date  at  which  any  debt  becomes 
due  for  the  standard  date,  and  find  how  long  after  that  date 
the  other  amounts  become  due. 

Find  the  average  term  of  credit  by  multiplying  each  debt  by 
the  number  of  days  from  the  standard  date,  and  dividing  the 
sum  of  the  products  by  the  sum  of  the  debts. 

Add  the  average  term  of  credit  to  the  standard  date,  and  the 
result  will  be  the  average  term  of  payment. 

Instead  of  the  earliest  date,  the  first  of  the  month  may  be  used. 

2.  What  is  the  average  time  at  which  the  following  bills 
become  due:  Feb.  1,  1887,  $200  on  1  mo.  credit;  March  10, 
1887,  $500  on  3  mo.  credit;  April  12,  1887,  $275  on  2  mo. 
credit;  and  May  1, 1887,  $400  on  4  mo.  credit? 


EXAMPLES.  371 

3.  A  merchant  owes  bills  dated  as  follows :  Jan.  1,  1892, 
$  500  due  in  2  mo. ;  Jan.  15,  1892,  $  850  due  in  3  mo. ;  Feb. 
20,  1892,  $375  due  in  3  mo. ;  and,  Feb.  28,  1892,  $650  due 
in  4  mo.     What  will  be  the  average  time  of  payment  ? 

4.  A  merchant  purchased  goods  of  Cragin  Bros.  &  Co.  as 
follows :  Sept.  10, 1891,  $  300  on  4  mo.  credit ;  Oct.  15, 1891, 
$400  on  6  mo.  credit;  Nov.  1,  1891,  $750  on  2  mo.  credit; 
and  Nov.  15,  1891,  $300  on  1  mo.  credit.     What  was  the 
average  time  of  payment  ? 

5.  Messrs.  J.  Eichards  &  Son  bought  goods  from  George 
C.  Buell  &  Co.  as  follows:  Sept.  1,  1891,  $600  on  3  mo. 
credit;  Oct.  3,  1891,  $400  on  4  mo.  credit;  Oct.  20,  1891, 
$250  on  2  mo.  credit;  and,  Nov.  10,  1891,  $375  on  1  mo. 
credit.     What  was  the  average  time  of  payment  ? 

6.  Stevens  &  Shepard   bought   goods  from  the  E-ussell 
Irwin  Manufacturing  Co.  as  follows :  Dec.  10,  1891,  a  bill 
of  $460  on  4  mo.  credit;  Jan.  5,  1892,  a  bill  of  $200  on  3 
mo.  credit;  Jan.  30,  1892,  a  bill  of  $200  on  4  mo.  credit; 
and,  Feb.  25,  a  bill  of  $  900  on  2  mo.  credit.     What  was  the 
average  time  of  payment  ? 

7.  Bought  goods  of  Carson,  Pirie  &  Co.  as  follows:  Jan. 
25, 1892,  $850  on  4  mo.  credit;  Feb.  15, 1892,  $ 600  on  3  mo. 
credit;  March  20,  1892,  $500  on  4  mo.  credit;  and,  April 
10,  1892,  $  960  on  2  mo.  credit.     What  was  the  average  time 
of  payment  ? 

8.  May  1,  1892,  Mr.  S.  purchased  goods  to  the  amount 
of  $  2400  on  the  following  terms :  \  payable  in  cash,  J  pay- 
able in  2  mo.,  and  the  balance  in  6  mo.     When  may  the 
whole  be  equitably  paid  by  one  payment  ? 

9.  A  bookseller  sold  $  1800  worth  of  books  upon  the  fol- 
lowing terms :  %  of  the  amount  in  cash,  $  600  payable  in  6 
months,  $  300  in  9  months,  and  the  rest  in  a  year.     What 
was  the  average  time  of  payment  ? 


AVERAGE   OF   ACCOUNTS. 


477.    1.    When  should  interest   begin  on   the   following 
account  ? 


Dr. 


H.  K.  GOODRICH. 


Cr. 


1892. 

1892. 

May 

5 

To  Mdse. 

50 

00 

May 

15 

By  Cash 

25 

00 

June 

7 

"       "      2  mo. 

140 

00 

June 

10 

"  Draft,  10  da. 

100 

00 

June 

21 

It            U           J     it 

150 

00 

June 

30 

U             (1 

100 

00 

PROCESS.     (By  Products.) 


Due, 

Amounti 

Days, 

Product. 

Paid, 

i 
Amount.  Days, 

Product, 

May 
Aug. 
July 

5 

7 
21 

$  50 
140 
150 

94 
0 
17 

4700 
2550 

May 
June 
June 

3150 

15 
23 

30 

-i-1 

$  25 
100 
100 

84 
45 
38 

WT 

2100 
4500 
3800 

340 
225 

7250 

225 
15  =  27 

10400 
7250 

115 

3150 

Aug.  7  +  27  days  =  Sept.  3,  the  average  time. 

EXPLANATION.  —  From  the  dates  at  which  the  various  amounts 
become  due,  we  select  the  latest,  which  is  Aug.  7,  for  the  assumed 
time  of  settlement,  and  multiply  each  amount  by  the  number  of  days 
intervening  between  that  date  and  the  time  when  each  item  of  the 
account  becomes  due.  The  debit  side  of  the  account  shows  there  is 
due  $  340  and  the  use  of  $  1  for  7250  days,  and  the  credit  side  shows 
that  $  225  has  been  paid,  and  that  the  debtor  is  entitled  to  the  use  of 
$1  for  10,400  days,  if  the  time  of  settlement  is  Aug.  7.  Subtracting 
the  amounts,  there  is  shown  to  be  $  115  due,  and  the  debtor  is  entitled 
to  the  use  of  $  1  for  3150  days.  Therefore,  he  should  not  be  required 

372 


EXAMPLES. 


373 


to  pay  the  account  until  the  time  when  the  use  of  $  115  is  equal  to  the 
use  of  $  1  for  3160  days,  which  is  27  days.  Hence,  interest  should 
begin  27  days  after  Aug.  7,  or  Sept.  3. 

2.   When    should    interest    begin   on   the    following   ac- 
count ? 

Dr.    JAMES  HOWARD,  in  acc't  with  HIRAM  SIBLEY.    Or. 


1892. 

1892. 

Apr. 

10 

To  Mdse. 

$150 

Apr. 

12 

By  Cash 

$250 

Apr. 

30 

u          tt 

400 

May 

1 

tl              tt 

200 

May 

16 

tt           it 

100 

June 

7 

u         tt 

400 

June 

24 

it              tt 

500 

3.    When  should  interest  begin  on  the  following  account  ? 
Dr.  S.  DANDRIDGE  &  Co.  Or. 


1892. 

1892. 

Jan. 

1 

To  Mdse.,  1  mo. 

$600 

Feb. 

3 

By  Cash 

$500 

Jan. 

20 

u         u         g    tt 

850 

Feb. 

28 

tt       tt 

200 

Feb. 

15 

u        u        2   " 

1500 

May 

15 

"   Draft,  1  mo. 

1200 

Apr. 

3 

it          tt          A     11 

2500 

4.  When  should  interest  begin  on  the  following  account  ? 
Dr.  WHITNEY  &  MYERS.  Or. 


1892. 

1892. 

Feb. 

1 

To  Mdse. 

$1800 

Feb. 

20 

By  Cash 

$3000 

Mar. 

15 

"       "      1  mo. 

3000 

May 

18 

"  Accept'  ce,  2  mo. 

8000 

Mar. 

20 

tt         tt        4     tt 

4800 

Apr. 

3 

tt           ti         A       tt 

6000 

5.    What  will  be   the   cash   balance    of    the    following 
account,  Jan.  1,  1892,  interest  at  6%  ? 

Dr.  HENRY  G.  SWINBURNE.  Or. 


1891. 

1891. 

July 

10 

To  Mdse.,  2  mo. 

$500 

July 

20 

By  Cash 

$400 

Aug. 

1 

u         u         3     u 

700 

Aug. 

20 

it       tt 

1000 

Sept. 

8 

U            It             1        '  ; 

800 

Sept. 

20 

tt       tt       2    u 

600 

SAYINGS  BANK  ACCOUNTS. 


478.  Savings    Banks   receive   small   sums  of  money  on 
deposit  and  pay  the  depositors  compound  interest,  either 
monthly,  quarterly,  or  semi-annually. 

479.  The  interval  between  the  dates  at  which  interest  is 
paid  is  called  an  Interest  Term. 

The  interest  term  in  most  banks  is  three  months,  beginning  Jan. 
1,  Apr.  1,  July  1,  and  Oct.  1.  Banks  whose  interest  term  is  one  month 
usually  begin  the  interest  term  on  the  first  day  of  each  month ;  those 
whose  term  is  six  months,  usually  on  Jan.  1  and  July  1. 

480.  Depositors  are  at  liberty  to  deposit  money  at  any 
time  and  generally  to  draw  it  out  at  their  pleasure,  but 
interest  is  computed  only  upon  the  sum  that  has  been  on 
deposit  during  the  whole  of  the  interest  term. 

481.  PRINCIPLE.  —  Interest  is  added  at  the  end  of  every  in- 
terest term  on  the  smallest  balance  on  deposit  during  the  entire 
term. 

1.  The  smallest  balance  on  deposit  at  any  time  during  the  term  is 
the  smallest  balance  on  deposit  during  the  entire  term. 

2.  In  examples  in  this  book  no  interest  is  computed  upon  the  cents 
in  the  balance,  and  in  the  interest  any  fractional  part  of  a  cent  is 
dropped.     However,  the  usage  of  banks  on  these  points  varies. 

482.  Depositors  in  savings  banks  are  given  a  bank-book 
in  which  all  deposits  and  amounts  drawn  out  are  recorded. 

WRITTEN  EXERCISES. 

483.  1 .   What  will  be  the  balance  due  on  the  following  ac- 
count on  Apr.  1, 1891,  interest  being  allowed  quarterly  at  4%  ? 

Deposited  Jan.  12,  1890,  $75;  May  10,  $150;  Sept.  1, 
$  20 ;  Jan.  1,  1891,  $  130. 

Drew  out  Mar.  5,  1890,  $30;  Aug.  16,  $50;  Dec.  1,  $48. 
374 


WRITTEN  EXERCISES. 


375 


STATEMENT. 


Bates, 

Deposited, 

Drew  out, 

Interest, 

Balance. 

1890 

Jan.  12 

75 

75 

Mar.    5 

30 

45 

Apr.    1 

00 

00 

45 

May  10 

150 

195 

July    1 

45 

195 

45 

Aug.  16 

50 

145 

45 

Sept.   1 

20 

165 

45 

Oct.     1 

1 

45 

166 

90 

Dec.    1 

48 

118 

90 

1891 

Jan.    1 

130 

1 

18 

250 

08 

Apr.    1 

2 

50 

252 

58 

EXPLANATION.  —  The  statement  is  an  exhibit  of  the  condition  of 
the  account  at  any  day  and  at  the  end  of  each  interest  term. 

At  the  end  of  the  first  interest  term,  no  interest  was  allowed  because 
no  part  of  the  deposits  was  in  the  bank  during  the  entire  term. 

The  smallest  balance  during  the  second  quarter  was  $45.  The 
interest  upon  it  ($  .45)  added  to  the  previous  balance  gives  the  balance 
July  1,  etc. 

The  balance  due  Apr.  1,  1891,  was  $252.58. 

Make  out  statements  and  find  the  balance  due  on  each  of 
the  following  accounts : 

2.  What  will  be  the  balance  due  on  the  following  account 
on  Jan.  1, 1891,  interest  being  allowed  semi-annually,  at  4%? 

Deposited  June  4,  1889,  $  175 ;  Nov.  1,  $  150 ;  Feb.  24, 
1890,  $200;  Sept.  10,  $56. 

Drew  out  Sept.  14,  1889,  $65;  July  25,  1890,  $120; 
Dec.  3,  $80. 

3.  What  will  be  the  balance  due  on  the  following  account 
on  Jan.  1,  1892,  interest  being  allowed  monthly  at  4%? 

Deposited  Jan.  1,  1890,  $36.50;  Mar.  17,  $25.38;  Aug. 
1,  $84.72;  June  11,  1891,  $50;  Nov.  16,  $40.78. 

Drew  out  Sept.  16,  1890,  $36.16;  Jan.  27,  1891,  $13.48; 
Mar.  1,  $  17.50. 


PBOGKESSIONS. 


484.  1.   How  does  each  of  the  numbers  2,  4,  6,  8,  10,  12, 
compare  with  the  number  that  follows  it  ? 

2.  How  may  each  of  the  numbers  4,  6,  8,  etc.,  be  obtained 
from  the  one  that  precedes  it  ? 

3.  How  does  each  of  the  numbers  2,  5,  8,  11,  14,  17,  com- 
pare with  the  number  that  follows  it  ?     How  with  the  one 
that  precedes  it  ? 

4.  Write  in  succession  some  numbers  beginning  with  3 
having  a  common  difference  of  2. 

5.  Write  a  series  of  numbers  beginning  at  4,  and  having 
a  common  difference  of  4. 

6.  Write  a  series  of  numbers  beginning  with  25,  and 
decreasing  by  the  common  difference  4. 

7.  How  does  each  of  the  numbers  2,  4,  8,  16,  32,  etc., 
compare  with  the  one  that  follows  it  ?     How  may  each  be 
obtained  from  the  one  that  precedes  it  ? 

8.  Write   a  series  of  numbers   beginning  with   2,  and 
increasing  by  a  common  multiplier  3. 

9.  Write   a  series  of  numbers   beginning  with   5,  and 
increasing  by  a  common  multiplier  5. 

485.  Numbers  in  succession,  each  derived  from  the  pre- 
ceding according  to  some  fixed  laws,  are  called  a  Series. 

486.  The  first  and  last  terms  of  a  series  are  called  the 
extremes,  the  intervening  terms  the  means. 

Thus,  in  the  series  2,  4,  6,  8,  10,  the  numbers  2  and  10  are  the 
extremes  and  the  others  are  the  means. 

376 


ARITHMETICAL  PROGRESSION.  377 

487.  A  series  in  which,  the  numbers  increase  regularly 
from  the  first  term  is  called  an  Ascending  Series. 

Thus,  2,  5,  8,  11,  14,  17,  20,  etc.,  is  an  ascending  series. 

488.  A  series  in  which  the  numbers  decrease  regularly 
from  the  first  term  is  called  a  Descending  Series. 

Thus,  48,  24,  12,  6,  3,  is  a  descending  series. 

ARITHMETICAL  PROGRESSION. 

489.  A  series  of  numbers  which  increase  or  decrease  by 
a  constant  common  difference  is  an  Arithmetical  Progression. 

Thus,  5,  9,  13,  17,  21,  etc.,  is  an  arithmetical  progression  of  which 
the  common  difference  is  4. 

490.  1.    The  first  term  of  an  arithmetical  series  is  3  and 
the  common  difference  is  2.     What  is  the  7th  term  ? 

PROCESS.  EXPLANATION.— Since 

Com.  diff.,   2  X  6    =  12  *he  comm°n  difference 

is  2,  the  second  term  is 

First  term,  3  -\- 12  =  15,  the  7th  term,     equal  to  the  first  plus 

once  the  common  dif- 
ference, the  third  term  is  equal  to  the  first  plus  twice  the  common 
difference,  the  fourth  term  is  equal  to  the  first  term  plus  three  times 
the  common  difference.  Hence,  the  seventh  term  will  be  equal  to  the 
first  term  plus  six  times  the  common  difference,  which  is  15. 

RULE. — Any  term  of  an  arithmetical  progression  is  equal 
to  the  first  term,  increased  or  diminished  by  the  common  differ- 
ence multiplied  by  a  number  one  less  than  the  number  of  terms. 

2.  A  boy  agreed  to  work  for  50  days  at  25  cents  the  first 
day,  and  an  increase  of  3  cents  per  day.     What  were  his 
wages  the  last  day  ? 

3.  A  body  falls  16^-  feet  the  first  second,  3  times  as  far 
the  second  second,  5  times  as  far  the  third  second.     How 
far  will  it  fall  the  seventh  second  ? 

4.  An  arithmetical  series  has  1000  terms,  the  first  term 
of  which  is  75  and  the  common  difference  5.     What  is  the 
last  term  ? 


378  PROGRESSIONS. 

5.  The  first  term  is  10  and  the  common  difference  5. 
What  is  the  10th  term  ?     Prove  it. 

6.  The  first  term  is  6  and  the  common  difference  is  8. 
What  is  the  25th  term  ? 

7.  Find  the  sum  of  an  arithmetical  series  of  which  the 
first  term  is  2,  the  common  difference  3,  and  the  number 
of  terms  7. 

2  +  (6  X  3)  =  20  EXPLANATION.  —  By  examining  the  series  2, 

2  4-  20  =  22  ^'  ®»  llj  14'  1^'  20'  Jt  is  evident  tliat  tlie  aver- 

911  a^e  term  is  11?  *or  if  kalf  tlae  sum  °*  anv  two 

22,  -TT    2  =  11  terms  equidistant  from  the  extremes  is  found, 

11  X  7  =  77  it  will  be  11,  and  in  general  in  any  arithmetical 
progression  the  average  term  is  equal  to  half  the 
sum  of  the  extremes  or  any  two  terms  equidistant  from  the  extremes. 
Since  the  first  term  is  2  and  the  common  difference  3,  the  last  term  is 
found  by  the  previous  rule  to  be  20.  The  sum  of  the  extremes  is 
therefore  22,  which,  divided  by  2,  gives  the  average  term.  And  since 
there  are  7  terms,  the  sum  will  be  7  times  the  average  term,  or  77. 

EULE.  —  To  find  the  sum  of  an  arithmetical  series :  Mul- 
tiply half  the  sum  of  the  extremes  by  the  number  of  terms. 

8.  What  is  the  sum  of  an  arithmetical  series  composed 
of  50  terms,  of  which  the  first  term  is  2  and  the  common 
difference  3  ? 

9.  What  is  the  sum  of  a  series  in  which  the  first  term  is 
^  the  common  difference  -j^,  and  the  number  of  terms  100  ? 

10.  A  man  walked  15  miles  the  first  day,  and  increased 
his  rate  3  miles  per  day  for  10  days.     How  far  did  he  walk 
in  the  eleven  days  ? 

11.  How  many  strokes  does  a  clock  strike  in  12  hours  ? 

12.  A  person  had  a  gift  of  $  100  per  year  from  his  birth 
until  he  became  21  years  old.     These  sums  were  deposited 
in  a  bank  and  drew  simple  interest  at  6%.     How  much  was 
due  him  when  he  became  of  age  ? 

13.  What  is  the  sum  of  the  series  in  which  the  first  term 
is  J,  the  common  difference  ^,  and  the  number  of  terms  100  ? 


GEOMETRICAL  PROGRESSION.  379 


GEOMETRICAL    PROGRESSION. 

491.  A  series  of  numbers  which  increase  or  decrease  by 
a  constant  multiplier  or  ratio  is  called  a  Geometrical  Pro- 
gression. 

Thus  5,  10,  20,  40,  80,  etc.,  is  a  geometric?!  progression,  of  which 
the  multiplier  or  ratio  is  2. 


WRITTEN    EXERCISES. 

492.  1.  The  first  term  of  a  geometrical  series  is  3  and 
the  multiplier  or  ratio  is  2.  What  is  the  5th  term  ? 

24  =  16  EXPLANATION.  — Since  the  multiplier  is  2,  the  sec- 

3   x  16  =  48     ond  term  wil1  be  3  x  2'  the  third  3  x  2  x  2  or  3  x  22, 
the  fourth  3  x  22  x  2  or  3  x  23,  and  the  fifth  3x 23x2 
or  3  x  24,  that  is,  the  fifth  term  is  equal  to  the  first  term  multiplied  by 
the  ratio  raised  to  the  fourth  power. 

RULE.  —  Any  term  of  a  geometrical  progression  is  equal  to 
the  first  term,  multiplied  by  the  ratio  raised  to  a  power  one  less 
than  the  number  of  the  term. 

2.  The  first  term  of  a  geometrical  progression  is  10,  and 
the  ratio  3.     What  is  the  6th  term? 

3.  The  first  term  of  a  geometrical  progression  is  10,  the 
ratio  4,  and  the  number  of  terms  6.    What  is  the  6th  term  ? 

4.  If  a  farmer  should  hire  a  man  for  10  days,  giving  him 
5  cents  for  the  first  day,  3  times  that  sum  for  the  second 
day,  and  so  on,  what  would  be  his  wages  for  the  last  day  ? 

5.  If  the  first  term  is  f  100  and  the  ratio  1.06,  what  is 
the  6th  term  ?     Or,  what  is  the  amount  of  $  100  at  com- 
pound interest  for  5  years  at  6%? 

6.  What  is  the  amount  of  $  520  for  6  years,  at  5%  com- 
pound interest? 


380  PROGRESSIONS. 

7.  What  is  the  sum  of  a  geometrical  series,  of  which  the 
first  term  is  5,  the  ratio  3,  and  the  number  of  terms  5? 
5  X  81=405,  the  5th  term.          EXPLANATION.  —  Since  in  this 
3  y  4-05       5  series  the  first  term  is  5,  the  ratio 

---  =  605,  the  sum.      3,  and  the  number  of  terms  5,  their 

sum  may  be  obtained  by  the  fol- 
lowing process,  which  illustrates  the  formation  of  the  rule: 

Series  5  +  15  +  45  +  135  +  405 
3  times  Series          15  +  45  +  135  +  405  +  1215 
2  times  Series      =  1215  —  6 


Series     =~  5 

EULE.  —  The  sum  of  a  geometrical  series  is  equal  to  the 
difference  between  the  first  term,  and  the  product  of  the  last 
term  by  the  ratio,  divided  by  the  difference  between  the  ratio 
and  1. 

Or,  since  the  last  term  is  equal  to  the  first  term,  multi- 
plied by  the  ratio  raised  to  a  power  one  less  than  the  num- 
ber of  terms, 

The  sum  of  a  geometrical  series  is  found  by  dividing  the 
difference  between  the  firsttterm  and  the  first  term  multiplied  by 
the  ratio  raised  to  the  power  equal  to  the  number  of  the  terms 
by  the  difference  between  the  ratio  and  1. 

8.  The  extremes  of  a  geometrical  progression  are  4  and 
1024,  and  the  ratio  is  4.     What  is  the  sum  of  the  series  ? 

9.  The  extremes  are  -J-  and  f|-|  and  the  ratio  is  2£. 
What  is  the  sum  of  the  series  ? 

10.  What  is  the  sum  of  the  series  in  which  the  first  term 
is  2,  the  last  term  0,  and  the  ratio  \  ;  or  what  is  the  sum  of 
the  infinite  series  2,  1,  ^  -J-,  -J-,  -j^,  -£-%,  etc.  ? 

11.  The  extremes  of  a  geometrical  progression  are  -^  and 
J^V2/,  and  the  ratio  is  1J.     What  is  the  sum  of  the  series  ? 

12.  A  man  put  money  in  the  bank  for  his  son  for  21 
years,  depositing  1  cent  the  first  year,  2  cents  the  second, 
4  cents  the  third,  etc.     How  much  did  he  deposit  in  all  ? 


PROBLEMS  IN  COMPOUND  INTEREST. 


WRITTEN   EXERCISES. 

493.  To  find  the  principal,  when  the  compound  interest,  the 
time,  and  the  rate  are  given. 

1.  What  principal  at  6%  compound  interest  will  produce 
$2372.544  interest  in  10  years  ? 

$  1.790848  —  $  1  =  $  .790848.  EXPLANATION.  —By  geomet- 

«  9379  *AA  •    7QAQJ.S       «  3000      rical  prograsswmj  or  by  the  com- 

°°-     pound  interest    table   on  page 

272,  the  amount  of  $  1  at  compound  interest  for  the  given  time  at  the 
given  rate  is  found  to  be  $  1. 790848.  That  sum  less  $  1  gives  $.  790848, 
the  compound  interest  of  $1  for  the  given  time  at  the  given  rate. 
Then,  $2372.544  -s-  .790848  =  $3000,  the  principal. 

2.  What  principal  at  6%  compound  interest  will  produce 
$  3150  interest  in  8  years  ? 

3.  What  principal  at  5%  compound  interest  will  produce 
$2896  interest  in  12  years  ? 

4.  What  principal  at  7%  compound  interest  will  produce 
$  3600  interest  in  15  years  ?     At  4%  in  20  years  ? 

494.  To  find  the  rate,  when  the  principal,  compound  interest 
and  time  are  given. 

1.    At  what  rate  per  cent  will  $500  yield  $203.55  com- 
pound interest  in  7  years  ? 

$  203.55 -J- 500  =  $.4071.         EXPLANATION.  — Since  $203.55  is  the 
compound  interest  of  $500  for  7  years, 

•sfar  of  that  sum  will  be  the  compound  interest  of  $1  for  the  same 
time.  By  referring  to  the  compound  interest  table,  opposite  7  years, 
we  find  the  amount  $  1.4071,  or  the  interest  $  .4071,  in  the  5%  column- 
Therefore,  the  rate  is  5%. 

381 


382         PROBLEMS  IN  COMPOUND  INTEREST. 

2.  At  what  rate  per  cent  will  $  1000  yield  $  503.63  com- 
pound interest  in  7  years  ? 

3.  At  what  rate  per  cent  will  $1200  yield  $721.2384 
compound  interest  in  12  years  ? 

4.  What  is  the  rate  per  cent  when  $  1800  yields  $  901.314 
compound  interest  in  6  years  ? 

5 .  What  is  the  rate  per  cent  when  $  2000  yields  $  4344.338 
compound  interest  in  15  years  ? 

495.   To  find  the  time  when  the  principal,  the  compound 
interest,  and  rate  are  given. 

1.  In  what  time  will  $600  amount  to  $1200  at  7%  com- 
pound interest  ? 

$  1200  -T-  600          =  $  2.  EXPLANATION.  —  Since  $  600 

$  2.  —  $  1.967151  =  $  .032849          amounts  to  $  1200  at  7  %  in  a 

« 1  Qfi71  W  v   07  —  «  1  37701  certain  time,  $  1  in  the  same 

*  1.967151  X  .07  -  ft  .137701         time  and  at  the  game  rate?  wm 

iVAVi  yr-  =  2  mo-  26  da.  amount  to  ^  of  $  1200,  or  $  2. 

.•.  The  time  =  10  yr.  2  mo.  26  da.         By  the  compound   interest 

table,  $1  at  7%  will  in  10  yr. 

amount  to  $1.967151,  and  in  11  yr.  to  $2.104852,  consequently  the 
time  must  be  between  10  and  11  yr.  The  interest  of  $1.967151  for  a 
year  at  7%  is  $.137701,  and  the  difference  between  $2  and  $1.967151 
is  $.032849.  Since  the  interest  of  $1.967151  for  a  year  is  $.137701, 
to  earn  $  .032849  will  require  fWrVr  of  a  year>  or  2  mo- 26  da-  There- 
fore the  time  is  10  yr.  2  mo.  26  da. 

2.  In  what  time  will  $400  amount  to  $1000,  at  6% 
compound  interest  ? 

3.  In  what  time  will  $750  amount  to  $1500,  at  5% 
compound  interest  ?     Or,  in  how  long  a  time  will  any  sum 
double  itself  at  5%  ? 

4.  In  what  time  will  $960  amount  to  $2000,  at  1% 
compound  interest  ? 

5.  In  what  time  will  $1300  amount  to  $2500,  at  6% 
compound  interest  ? 

6.  In  what  time  will  $3200  amount  to  $4800,  at  4% 
compound  interest  ? 


ANNUITIES. 


496.  A  definite  sum  of  money  payable  at  the  end  of  equal 
periods  of  time  is  an  Annuity. 

Properly  speaking,  an  annuity  is  a  sum  payable  annually,  but  sums 
payable  at  intervals  of  quarter-years,  half-years,  or  other  periods,  are 
also  called  annuities. 

497.  An  annuity  which  continues  forever  is  called  a  Per- 
petual Annuity  or  Perpetuity. 

498.  An  annuity  which  commences  at  a  definite  time, 
and  continues  for  a  definite  time,  is  called  a  Certain  Annuity. 

499.  An  annuity  whose  commencement  or  continuance, 
or  both,  depend  upon  some  contingent  event,  as  the  death 
of  some  person,  is  called  a  Contingent  Annuity. 

500.  An  annuity  upon  which  the   payments  were   not 
made  when  they  were  due  is  called  an  Annuity  in  Arrears  or 
Forborne. 

501.  The  Amount  or  Final  Value  of  an  annuity  is  the  sum 
of  all  the  payments,  increased  by  the  interest  of  each  pay- 
ment, from  the  time  it  becomes  due  until  the  annuity  ceases. 

502.  A  sum  of  money,  which,  upon  being  put  at  interest 
for  the  given  time  at  the  given  rate,  will  be  equal  to  the 
amount  of  the  annuity,  is  the  Present  Value  of  the  annuity. 

503.  Annuities  are  sometimes  computed  at  Simple  Inter- 
est and  sometimes  at  Compound  Interest. 


384  ANNUITIES. 

WRITTEN   EXERCISES. 
504.   Annuities  at  Simple  Interest. 

1.  What  is  the  amount  of  an  annuity  of  $  500,  unpaid  for 
5  yr.,  at  6%  ? 

I  =  $  500  +  $    30x4=$    620  EXPLANATION.  —  Since  the  an- 

$500+1620   K  (tt9ftnn    nuity  was  unPai*  f°r  5  y-'  the 

s=-  -  i  -  XO=§>Z800       iirst  payment  will  draw  interest 
^  from  the  end  of  the  first  year  un- 

til the  time  of  payment,  or  4  yr.  ;  the  second  payment  will  draw  inter- 
est for  3  yr.  ;  the  third  payment,  for  2  yr.  ;  the  fourth  payment,  for 
1  yr.  Hence,  these  sums  form  an  arithmetical  progression,  the  first 
term  of  which  is  $  500,  the  common  difference,  the  interest  of  $  500  for 
1  yr.,  or  $30,  and  the  number  of  terms,  5.  The  sum  of  this  series  will 
be  the  amount  due. 

2.  What  is  the  present  worth  of  an  annuity  of  $2500  to 
remain  unpaid  for  6  years,  interest  at  6%  ? 


$  17,250  =  Amount  of  annuity. 

$  17,250  -*-  1.36  =  $  12,683.82        be  $  17,250. 

Since  this  sum  is  not  payable 

for  6  years,  the  present  worth  of  it  is  found  by  dividing  by  the  amount 
of  $  1  for  the  given  time  at  the  given  rate. 

3.  What  is  the  amount  of  an  annuity  of  $800,  unpaid 
for  4  years,  at  6%  ? 

4.  What  is  the  amount  of  an  annuity  of  $960,  payable 
semi-annually,  at  6%,  but  unpaid  for  4  years  ? 

5.  What  is  the  present  worth  of  an  annuity  of  $1500,  to 
remain  unpaid  for  8  years,  at  6%  ? 

6.  Mr.  L.  has  an  annuity  of  $  1800,  payable  quarterly. 
If  it  remains  unpaid  for  3  years  9  months,  what  will  be  the 
amount  due  at  8  %  ? 

7.  A  house  was  rented  for  $45  a  month  for  2|-  years. 
What  sum  would  pay  the  entire  rent  in  advance  if  it  was 
not  due  until  the  lease  expired,  interest  at  6%  ? 


ANNUITIES  AT   COMPOUND  INTEREST.         385 

505.   Annuities  at  Compound  Interest. 
1.    What  is  the  amount  of  an  annuity  of  $200  which  is 
20  years  in  arrears,  compound  interest  at  6%  ? 

EXPLANATION.  —  The  payment 

$200  X  (1.0620— l)_fl 7oK7  -jo     now  due  is  $200  ;  the  payment  1 
1.06—1  year  in  arrears  is  $200  x  1.06 ;  the 

payment  2  years  in  arrears  is  $200 

x  1.06  x  1.06,or$200  x  1.062;  the  payment  3  years  in  arrears  is  $200  x  1.063. 
Thus  it  appears  that  the  sums  unpaid  form  a  geometrical  series,  of 
which  the  first  term  is  $200,  the  ratio  1.06,  and  the  number  of  terms 
20.    The  sum  of  this  series  is  $  7357.12,  the  amount  of  the  annuity. 

1.  In  finding  the  value  of  1.0620,  or  similar  expressions,  use  the 
compound  interest  table  on  page  272. 

2.  The  amount  of  an  annuity  at  simple  interest  is  the  sum  of  an 
arithmetical  series  ;  the  amount  of  an  annuity  at  compound  interest  is 
the  sum  of  a  geometrical  series. 

2.  What  is  the  amount  of  an  annuity  of  $  225,  which  is 
6  years  in  arrears,  compound  interest  at  1%  ? 

3.  What  is  the  amount  of  an  annuity  of  $300,  which  is 
9  years  in  arrears,  compound  interest  at  6%  ? 

4.  What  is  the  amount  of  an  annuity  of  $450,  in  arrears 
for  15  years,  compound  interest  at  5%  ? 

5.  What  is  the  amount  of  an  annuity  of  $  650,  in  arrears 
for  10  years,  the  interest  being  compounded  semi-annually, 
at  6%  ? 

6.  A  young  man  spends  $50  a  year  for  tobacco.     What 
will  this  amount  to  in  20  years,  at  6%  compound  interest? 

7.  What  is  the  present  value  of  an  annuity  of  $800  for 
6  years,  compound  interest  at  5%  ? 

SUGGESTION.  —  Divide  the  amount  of  the  annuity  by  the  amount  of 
$1,  at  compound  interest,  for  the  given  time  and  rate. 

8.  WTiat  is  the  present  worth  of  an  annuity  of  $480  for 
12  years,  compound  interest  at  6%  ? 

9.  A  man  purchased  an  annuity  of  $600  a  year  for  15 
years,  at  6%  compound  interest.     What  did  it  cost  him  ? 

STAND.    AR. — 25 


DIVISORS   AND   MULTIPLES. 


COMMON  DIVISORS. 

506.  1.   What  numbers  will  exactly  divide  12  ?   15?   20? 

2.  What  numbers  will  exactly  divide  both  12  and  15  ? 
15  and  20  ?     24  and  48  ?     63  and  72  ? 

3.  What  numbers  will  exactly  divide  both  12  and  24  ? 
What  is  the  largest  number  that  will  exactly  divide  them  ? 

4.  What  is  the  largest  number  that  will  exactly  divide 
both  15  and  30?     16  and  32?     16  and  24?    24  and  32? 

5.  Name  all  the  divisors  common  to  15  and  30. 

6.  Name  all  the  prime  divisors  or  factors  common  to  15 
and  30  ? 

7.  How   is   the   greatest  divisor  common  to  15  and  30 
found  from  the  prime  factors  of  those  numbers  ? 

8.  What  is  the  greatest  divisor  common  to  24  and  30? 

9.  How   is   the    greatest  divisor  common  to  24  and  30 
obtained  from  the  prime  factors  of  those  numbers  ? 

507.  A  number  that  is  an  exact  divisor  of  two  or  more 
numbers  is  called  a  Common  Divisor  of  the  numbers. 

508.  The  greatest  number  that  is  an  exact  divisor  of  two 
or  more  numbers  is  called  the  Greatest  Common  Divisor  of 
the  numbers. 

Thus  12  is  the  greatest  common  divisor  of  12  and  24. 


COMMON  DIVISORS.  387 

509.  PRINCIPLE.  —  The  greatest  common  divisor  of  two  or 
more  numbers  is  the  product  of  all  their  common  prime  factors. 

WRITTEN   EXERCISES. 

510.  1.    What  is  the  greatest  common  divisor  of  42,  63, 
and  126? 

A  9       9       Q       7  EXPLANATION.  —  Since  the  greatest  com- 

mon divisor  is  equal  to  the  product  of  all  the 

63  =  3  X  3  X  7  prime  factors  common  to  the  given  numbers, 

9  Q  Q  7  we  sePara^e  the  numbers  into  their  prime 
=JxoXoX7  factors.  The  only  common  prime  factors 
3  X  7  =  21.  are  3  and  7  ;  hence,  their  product,  21,  is  the 

greatest  common  divisor  of  the  given  num- 
bers. 
42     63    126  The  common  prime  factors  may  be  found 


6       9       18  readily  by  dividing  the  numbers  by  the  prime 

"o       o         g  factors  successively,  until  the  quotients  con- 

tain no  common  factor. 

2.  24,120.  8.  135,225.  14.  36,42,54. 

3.  33,  154.  9.  232,  493.  15.  48,  60,  96. 

4.  42,  252.  10.  210,  350.  16.  120,  210,  345. 

5.  60,  270.  11.  330,  495.  17.  216,  360,  432. 

6.  56,  126.  12.  352,  384.  18.  42,  63,  126,  189. 

7.  112,168.  13.  840,1260.  19.  126,210,294,462. 

511.   Sometimes  the  numbers  cannot  be  readily  factored. 
In  such  cases  the  following  method  is  employed : 

1.    What  is  the  greatest  common  divisor  of  35  and  168  ? 

35)168(4  EXPLANATION.  —  The  greatest  common  divisor 

+  AQ  cannot  be  greater  than  the  smaller  number ;  there- 

fore  35  will  be  the  greatest  common  divisor  if  it 

28)35(1          is  exactly  contained  in  168.     By  trial  it  is  found 
28  that  it  is  not  an  exact  divisor  of  168,  since  there 

~7\28(4  is  a  remainder  of  28.      Therefore  35  is  not  the 
9£        greatest  common  divisor. 

Since  168  and  140,  which  is  4  times  35,  are 

each  divisible  by  the  greatest  common  divisor 

(Art.  106,  11),  their  difference,  28,  must  be  divisible  by  the  greatest 


388  DIVISORS  AND  MULTIPLES. 

common  divisor,  therefore  the  greatest  common  divisor  cannot  be 
greater  than  28.  28  will  be  the  greatest  common  divisor  if  it  is  exactly 
contained  in  35 ;  since  if  it  is  contained  in  35,  it  will  be  contained  in 
140  (Art.  106,  11),  and  in  28  plus  140,  or  168.  By  trial  we  find  that 
it  is  not  an  exact  divisor  of  35,  for  there  is  a  remainder  of  7.  There- 
fore 28  is  not  the  greatest  common  divisor. 

Since  28  and  35  are  each  divisible  by  the  greatest  common  divisor, 
their  difference,  7,  must  contain  the  greatest  common  divisor  (Art. 
106,  12);  therefore  the  greatest  common  divisor  cannot  be  greater 
than  7.  7  will  be  the  greatest  common  divisor  if  it  is  exactly  con- 
tained in  28 ;  since  if  it  is  contained  in  itself  and  28,  it  will  be  con- 
tained in  the  sum,  35,  and  also  in  168,  which  is  the  sum  of  28  and  4 
times  35,  or  140.  By  trial  we  find  that  it  is  an  exact  divisor  of  28. 
Hence  7  is  the  greatest  common  divisor. 

RULE.  —  Divide  the  greater  number  by  the  less,  and  if  there 
is  a  remainder,  divide  the  less  number  by  it,  then  the  preceding 
divisor  by  the  last  remainder,  and  so  on,  till  nothing  remains. 
The  last  divisor  will  be  the  greatest  common  divisor. 

If  more  than  two  numbers  are  given,  find  the  greatest 
common  divisor  of  any  two,  then  of  this  divisor  and  another 
of  the  given  numbers,  and  so  on.  The  last  divisor  will  be  the 
greatest  common  divisor. 

Find  the  greatest  common  divisor  of 

2.  252,  280.           5.    756,  1575.             8.  146,  365,  219. 

3.  323,  425.           6.    1008,  1036.           9.  225,  315,  420. 

4.  432,  936.           7.    1088,  1632.         10.  462,  882,  546. 

Reduce  the  following  fractions  to  their  lowest  terms  by 
finding  the  greatest  common  divisor  of  their  terms : 

11.  AV        ".  f|f.        17.  Hf        20. 

12.    ftf  15.    ||f.  18.    fff  21. 

13-    ttf  16.    fff  19.    flyj.  22. 

23.  A  farmer  wishes  to  put  336  bushels  of  wheat  and 
576  bushels  of  corn  into  the  least  number  of  bins  possible 
of  uniform  size,  without  mixing  the  two  kinds  of  grain. 
How  many  bushels  must  each  bin  hold  ? 


COMMON  MULTIPLES.  389 


COMMON  MULTIPLES. 

512.  1.    Name  some  numbers  that  are  exactly  divisible 
by  2.     By  3.     By  4.     By  2,  3,  and  4. 

2.  What  is  the  smallest  number  that  is  exactly  divisible 
by  each  of  the  numbers  2,  3,  and  4  ? 

3.  What  is  the  least  number  that  will  contain  10  and  15  ? 

4.  What  common  prime  factors  have  10  and  15?    What 
factor  occurs  in  10  that  does  not  occur  in  15  ?     What  fac- 
tor is  found  in  15  that  is  not  found  in  10  ? 

5.  What  are  all  the  different  prime  factors  of  10  and  15  ? 

6.  How  may  the  least  number  that  will  contain  10  and 
15  be  formed  from  their  prime  factors  ? 

What  is  the  least  number  that  will  exactly  contain 

7.  3,  6,  and  9  ?  12.    2,  3,  5,  and  6  ? 

8.  3,  5,  and  6  ?  13.    3,  4,  5,  and  6  ? 

9.  4,  8,  and  12  ?  14.    3,  6,  8,  and  12  ? 

10.  7,  14,  and  28  ?  15.    4,  8,  12,  and  15  ? 

11.  4,  6,  and  10  ?  16.    5,  6,  10,  and  12  ? 

513.  A  number  that  will  exactly  contain  another  number 
is  a  Multiple  of  that  number. 

514.  A  number  that  will  exactly  contain  each  of  two  or 
more  numbers  is  a  Common  Multiple  of  those  numbers. 

515.  The  least  number  that  will  exactly  contain  each  of 
two  or  more  numbers  is  their  Least  Common  Multiple. 

516.  PRINCIPLE.  —  The  least  common  multiple  of  two  or 
more  numbers  is   equal  to  the  product  of  all  their  different 
prime  factors,  each  factor  used  the  greatest  number  of  times 
that  it  occurs  in  any  of  the  numbers. 


390  DIVISORS  AND  MULTIPLES. 

WRITTEN   EXERCISES. 
517.   1.    Find  the  least  common  multiple  of  30,  28,  and  60. 

EXPLANATION.— To  apply  the  princi- 

30  =  2  X  3  X  5  pie  of  Art.  516,  all  the  numbers  must  be 

9g  _  2  v  2  X  7  separated  into  their  prime  factors.   Then, 

*^  the  factors  of  the  least  common  multiple 

=  J  X  2  X  6  X  5  are  2,  2   (the   greatest  number  of  2's 

2x2x3x5x  7=420       found  in  any  number)  and  3,  5,  7  (the 

only  prime  factors  of  any  of  them  not 

already  taken).     Therefore,  the  product  of  2,  2,  3,  5,  and  7  is  the 
least  common  multiple  of  the  numbers. 


30        28        60  EXPLANATION.  —  By  dividing   the   given 


15        14       30  numbers  by  any  prime   number  that  will 

^jj~          ir        7^  exactly  divide  two  or  more  of  them,  until 

— : — quotients  are  found  that  are  prime  to  each 

_^ '  ^  other,  and  then  finding  the  product  of  these 

171  divisors  and  the   last  quotients,    the  least 

2x2x3x5x  7  =  420  common  multiple  of  the  numbers  is  found. 

In  finding  the  least  common  multiple  of  numbers,  all  numbers  that 
are  factors  of  other  given  numbers,  may  be  disregarded. 

Thus,  the  common  multiples  of  4,  8,  16,  32,  64,  80,  and  128  are  the 
same  as  the  common  multiples  of  80  and  128. 

Find  the  least  common  multiple  of 

2.  16,  20,  48,  60.  9.  126,  36,  48,  66. 

3.  18,  21,  27,  36.  10.  16,  60,  140,  210. 

4.  20,  35,  40,  45.  11.  57,  36,  231,  330. 

5.  36,  40,  48,  126.  12.  126,  140,  154,  280. 

6.  7,  11,  91,  13.  13.  48,  117,  54,  312. 

7.  45,  75,  135,  180.  14.  63,  72,  84,  105. 

8.  96,  126,  72,  56.  15.  132,  144,  288,  324. 

518.  When  the  prime  factors  of  the  given  numbers  can- 
not be  discovered  by  inspection,  they  may  be  found  by  the 
method  of  finding  the  greatest  common  divisor  under  such 
circumstances. 


COMMON  MULTIVLES.  391 

16.  Find  the  least  common  multiple  of  255  and  357* 

255)357(1 
255 

102") 255 (2  EXPLANATION.  — Since  the  factors  of  the 

204-  numbers  cannot  be  discovered  by  inspec- 

•  tion,  the  greatest  common  divisor  of  the 

51 )  102  (2  numbers  is  found  to  be  51 . 

Dividing  each  of  the  given  numbers  by  51, 

.  the  quotients  5  and  7  are  obtained  which 

.-.  Ine  Cr.  O.  L>.  is  51.  are  prime  to  each  other     Therefore  51  x  5 

51)255  357  x  7>  or  1785,  is  the  least  common  multiple 

K ~  of  the  numbers. 

o       i 

51  x  5  x  7  =  1785. 
Find  the  least  common  multiple  of 

17.  315,  420.  20.    468,  923.  23.    777,  1110. 

18.  448,  512.  21.   432,  936.  24.   2310,  3150. 

19.  560,  616.  22.    720,  868.  25.   2520,  2772. 

26.  Find  the  contents  of  the  smallest  vessel  that  may  be 
filled  by  using  a  3-quart,  a  4-quart,  a  5-quart,  or  a  6-quart 
measure. 

27.  What  is  the  shortest  length  that  can  be  measured  by 
either  of  four  measures  which  are  respectively  10  in.,  15 
in.,  27  in.,  and  30  in.  long  ? 

28.  A  can  walk  round  a  race-course  in  12  min.,  B  in  15 
min.,  and  C  in  18  min.     If  they  start  together  and  keep 
walking   each   at  his  own   rate,  how  many  minutes  will 
elapse  before  they  are  all  three  together  at  the  starting- 
point,    and   how   many   times   will    each   have    made    the 
circuit  ? 

29.  A  lady  desires  to  purchase  a  quantity  of  cloth  that 
can  be  cut  without  waste  into  parts  4,  5,  or  6  yards  long. 
What  is  the  least  number  of  yards  that  she  can  buy  for 
that  purpose  ? 


392  DIVISORS   AND   MULTIPLES. 


GREATEST  COMMON  DIVISOR  OP  FRACTIONS. 

519.  1.    Give  several  fractions  which  are  contained  in  f 
an  integral  number  of  times. 

2.  What  relation  do  the  numerators  of  these  divisors 
bear  to  the  numerator  of  |-  ? 

3.  What  relation  do  the  denominators  of  these  divisors 
bear  to  the  denominator  of  f  ? 

4.  Find  several  fractions  which  are  common  divisors  of 
fandf.     Of  |  and  f.     Of  f  and  T3¥.     Of  f  and  f  . 

5.  What  relation  do  the  numerators  of  these  common  divi- 
sors bear  to  the  numerators  of  the  fractions  which  they  divide? 

6.  What  relation  do  the  denominators  of  these  common 
divisors  bear  to  the  denominators  of  the  fractions  ? 

7.  Since  the  fraction  which  will  exactly  divide  the  given 
fractions  is  greatest  when  its  numerator  is  as  large  as  pos- 
sible, and  its  denominator  as  small  as  possible,  how  are  the 
terms  of  the  greatest  common  divisor  of  fractions  obtained 
from  the  terms  of  the  fractions  ? 

520.  PRINCIPLE.  —  The  greatest  common  divisor  of  two  or 
more  fractions  in  their  lowest  terms  is  the  greatest  common 
divisor  of  their  numerators  divided  by  the  least  common  multi- 
ple of  their  denominators. 

WRITTEN   EXERCISES. 

521.  Find  the  greatest  common  divisor  of  : 
I-    iff  3.    |f  T<A,|f  5.    4 


2-    ft  If  if  4.    3f  If  |f  6.    SfftfJ 

7.  The  sides  of  a  triangular  lot  are  115^  feet,  128^  feet, 
and  134J  feet  long.  How  many  rails  of  the  greatest  length 
possible  will  be  needed  to  fence  it,  the  rails  lapping  6  inches 
at  each  end,  and  the  fence  to  be  7  rails  high  ? 


L.C.M.   OF  FRACTIONS.  393 


LEAST  COMMON  MULTIPLE  OP   FRACTIONS. 

522.  1.    Give  several  fractions  or  integers  which  will 
contain  i-  an  integral  number  of  times  ^,  T^-,  -|,  J,  -fa. 

2.  What  relation  do  the  numerators  of  these  multiples 
bear  to  the  numerators  of  the  given  fractions  ? 

3.  What  relation  do  the  denominators  of  these  multiples 
bear  to  the  denominators  of  the  given  fractions  ? 

4.  Find  several  common  multiples  of  J-  and  -J.     Of  f  and 
f .     Of  i  and  }. 

5.  What  relation  do  the  numerators  of  these  common 
multiples  bear  to  the  numerators  of  the  fractions  which 
they  contain  ? 

6.  What  relation  do  the  denominators  of  these  common 
multiples  bear  to  the  denominators  of  the  fractions  which 
they  contain  ? 

7.  Since  the  number  that  will  exactly  contain  the  given 
fractions  is  least  when  its  numerator  is  as  small  as  possible 
and  its  denominator  as  large  as  possible,  how  are  the  terms 
of  the  least  common  multiple  of  fractions  obtained  from  the 
terms  of  the  fractions  ? 

523.  PRINCIPLE.  —  The  least  common  multiple  of  two  or 
more  fractions  in  their  lowest  terms  is  the  least  common  multi- 
ple of  their  numerators  divided  by  the  greatest  common  divisor 
of  their  denominators. 

WRITTEN   EXERCISES. 

524.  Find  the  least  common  multiple  of : 

i-  i>  f>  *•          3-  A>  A* »        5-  T%>  iffc 

2-   A>  H>  H-          4.    2|,  3|,  4^.         6.    11^,  142, 
7.    The  pendulum  of  one  clock  makes  25  beats  in  28 
seconds,  and  that  of  another  clock  30  beats  in  34  seconds. 
If  the  clocks  are  started  at  the  same  moment,  when  first 
after  starting  will  the  clocks  beat  together  again  ? 


CIECULATING   DECIMALS. 


525.  1.    When  a  cipher   is   annexed   to   a  number,   by 
what  is  the  number  multiplied  ?     After  the  cipher  has  been, 
annexed,  by  what  numbers  can  the  number  be  divided  by 
which  it  could  not  be  divided  before  ? 

2.  Since  the  only  new  factors  by  which  a  number  multi- 
plied by  10  can  be  divided  are  2  and  5,  when  a  common 
fraction  in  its  lowest  terms  is  being  reduced  to  a  decimal, 
if  the  denominator  contains  only  the  factors  2  or  5,  will  the 
division  be  exact  or  not  ? 

3.  If  the  denominator  contains  other  factors  besides  2  or 
5,  what  can  be  said  of  the  division  ? 

4.  If  any  fraction,  as  -f,  is  reduced  to  a  decimal  by  an- 
nexing ciphers  to  the  numerator  and  dividing  by  the  denom- 
inator, how  many  possible  remainders  can  there  be  ? 

5.  Since  in  each  instance  the  remainder  with  a  cipher 
annexed  forms  the  new  dividend,  and  since  there  can  be 
but  6  different  dividends,  what  may  be  concluded  regarding 
the  repetition  of  the  decimal  figures  ? 

6.  What,  then,  may  be  inferred  regarding  the  repetition 
of  the  decimal  figures  in  any  infinite  decimal  ? 

526.  A  decimal  that  contains  a  definite  number  of  deci- 
mal places  is  called  a  Finite  Decimal.     A  decimal  that  never 
terminates  is  called  an  Infinite  Decimal. 

527.  An  infinite  decimal  having  a  figure  or  set  of  figures 
repeated  indefinitely  is  called  a  Circulating  Decimal. 

394 


DEFINITIONS.  395 

528.  The  figure  or  set  of  figures  repeated  in  an  infinite 
or  circulating  decimal  is  called  the  Repetend. 

Thus,  the  common  fraction  £  is  expressed  by  .3333  +  etc. ;  the  frac- 
tion |  by.142857142857  +  etc.  In  the  first  fraction  the  repetend  is  3  ; 
in  the  second  142857. 

A  repetend  is  indicated  by  placing  a  dot  over  the  repeated  figure  ; 
or  over  the  first  and  last  figures  of  the  set  that  is  repeated. 

Thus,  .333  +  is  written  .3 ;  .142857142857  +  is  written  .142857  ; 
,1666+  is  written  .16. 

529.  A  decimal  expressed  wholly  by  a  repetend  is  called 
a  Pure  Circulating  Decimal. 

Thus,  .3333  +  and  .142857142857  +  are  pure  circulating  decimals. 

*    530.   A  decimal  expressed  only  in  part  by  a  repetend  is 
called  a  Mixed  Circulating  Decimal. 

Thus,  .1666  +  etc.,  .4535353  +  are  mixed  circulating  decimals. 

531.  PRINCIPLES.  —  1.    Any  fraction  in  its  lowest  terms 
whose  denominator  contains  no  other  prime  factors  besides  2 
or  5  can  be  reduced  to  a  finite  decimal. 

2.  Any  fraction  in  its  lowest  terms  whose  denominator  con- 
tains other  prime  factors  besides  2  or  5  will  produce  a  circu- 
lating decimal. 

EXERCISES. 

532.  Tell  by  inspection  which   fractions   will   produce 
finite  decimals  and  which  circulating  decimals. 

1.  J.  4.    ^.  7.    |  10.    f.  13.    ^. 

2.  f.  5.     f  8.    f  11.    |.  14.    A- 
3-    f.            6.     f             9.    f.            12.    f            15.    A- 

16.    Reduce  \  to  a  decimal.     f .     f .     -J. 

17     Keduce  -£$  to  a  decimal.     -^-.     ff.     £J-. 

18.    Eeduce  -       to  a  decimal. 


396  CIRCULATING  DECIMALS. 

533.  To  reduce  a  repetend  to  a  common  fraction. 

1.  To  what   common   fraction   is   .1111+   or  .1  equal? 
.5555+  or  .5  ?     .7777+  or  .7  ?     (Art.  532,  Ex.  16.) 

2.  To  what  common  fraction  is  .010101+  or  .61  equal? 
.050505+  or  .05  ?     .373737+  or  .37  ? 

3.  To  what  common  fraction  is  .001001+  or  .OOl  equal  ? 
.007007+  or  .007  ?     .356356+  or  .356  ? 

534.  PRINCIPLE.  —  The  denominator  of  a  pure  circulating 
decimal  is  as  many  9's  as  there  are  figures  in  the  repetend. 

EXERCISES. 

535.  Express  as  common  fractions  : 

1.  .3.  4.    .324.  7.    .642.  10.   .3636. 

2.  .14.  5.    .378.  8.    .963.  11.    .00261. 

3.  .123.  6.    .045.  9.    .9801.  12.    .986013. 

13.   Express  as  common  fractions,  .0635  and  .5347. 


EXPLANATION.  —  If  the  decimal  were  .635,  its  value  as  a  common 
fraction  would  be  f  f  f  .  But  since  the  repetend  begins  one  place  to  the 
right  of  the  decimal  point,  its  value  is  T^  of  fff.  Therefore  .0635 


.5347 

Express  as  common  fractions  : 

14.  .165.         17.    .5635.         20.    .0815.         23.    .09563. 

15.  .256.         18.    .2045.         21.    .5622.         24.    .009867. 

16.  .327.         19.    .3572.         22.    .3156.         25.    .985375. 


SCALES   OF   NOTATION. 


536.  The  ratio  by  which,  numbers  increase  or  decrease  is 
termed  a  Scale. 

The  ordinary  scale  of  notation  for  integers  is  decimal,  but  it  is 
possible  to  express  numbers  in  many  other  scales. 

537.  The  number  of  units  required  to  make  one  of  the 
next  higher  order  is  called  the  Radix  of  the  scale. 

Thus  10  is  the  radix  of  the  decimal  scale,  12  of  the  duodecimal. 

SCALES. 


Name. 

Radix. 

Name. 

Radix. 

Binary 

2 

Septenary 

7 

Ternary 

3 

Octary 

8 

Quaternary 

4 

Nonary 

9 

Quinary 

5 

Decimal 

10 

Senary 

6 

Undenary 

11 

Duodecimal 

12 

NOTATION. 

538.  In  expressing  numbers  in  any   uniform  scale,   as 
many  characters  must  be  employed  as  there  are  units  in 
the  radix  of  the  scale,  and  one  of  the  characters  must  be  0. 

To  express  numbers  in  scales  higher  than  the  decimal,  new  char- 
acters must  be  employed.  Thus,  t  may  be  used  to  represent  10  and 
e  to  represent  11,  etc. 

539.  Inasmuch  as  the  names  of  the  orders  of  units  used 
in  expressing  numbers  are  adapted  to  the  decimal  scale, 

397 


898  SCALES  OF  NOTATION. 

numbers  in  other  scales  should   be  read  by  naming  the 
number  of  units  of  each  order. 

Thus,  342  in  the  quinary  scale  should  be  read :  quinary  scale,  3  units 
of  the  third  order,  4  of  the  second,  and  2  of  the  first. 

WRITTEN   EXERCISES. 

540.  1.   Write  in  the  quinary  scale  the  numbers  .corre- 
sponding to  the  numbers  from  1  to  13  in  the  common  or 
decimal  scale. 

EXPLANATION. — Since  the  radix  of  the  scale  is  5,  the  characters 
employed  are  1,  2,  3,  4,  0. 

Since  5  units  of  any  order  are  equal  to  1  of  the  next  higher  order, 
the  numbers  including  5  will  be  expressed  by  1,  2,  3,  4,  10. 

Since  6  is  equal  to  1  unit  of  the  second  order  and  1  of  the  first  order 
it  is  written  11 ;  since  7  is  equal  to  1  unit  of  the  second  order  and  2  of 
the  first  it  is  written  12. 

Expressing  the  numbers  from  1  to  13  in  accordance  with  the  law 
just  illustrated,  they  are  1,  2,  3,  4,  10,  11,  12,  13,  14,  20,  21,  22,  23. 

Write  the  numbers  corresponding  to  the  numbers  from  1 
to  20  in  the  common  or  decimal  scale : 

2.  In  the  quaternary  scale.       6.    In  the  nonary  scale. 

3.  In  the  octary  scale.  7.    In  the  undenary  scale. 

4.  In  the  senary  scale.  8.   In  the  septenary  scale. 

5.  In  the  ternary  scale.  9.    In  the  binary  scale. 

541.  To  change  from  the  decimal  to  another  scale. 

1.   Change  58375  from  the  decimal  to  the  senary  scale. 


58375  EXPLANATION.  —  By  dividing  by  6,  we  obtain  the 


9729  -4-  1     number  of  units  of  the  second  order  and  the  number 
of  units  of  the  first  order  remaining. 

By  continuing  to  divide  by  6,  the  number  of  units 


+  3 


270  +  1     in  the  successive  orders  is  obtained  and  the  number 
45  _j_  0     °f  units  remaining  after  division.    It  is  thus  found 


7  +  3 


that  58375  when  expressed  in  the  senary  scale  con- 


tains 1  unit  of  the  seventh  order,  1  of  the  sixth,  3  of 
1  4- 1     the  fifth,  etc. ,  or  the  number  is  expressed  by  11301316. 
For  convenience  in  notation  the  radix  of  the  scale  is  indicated  by  a 
small  subscript  figure. 


WRITTEN  EXERCISES.  399 

2.  Express  in  the  quinary  scale,  3824,  5861,  and  3843. 

3.  Express  in  the  septenary  scale,  5163,  6842,  and  4276. 

4.  Express  in  the  quaternary  scale,  3947,  5439,  and  3854. 

5.  Express  in  the  duodecimal  scale,  6193,  8*27,  and  6958. 

542.  To  change  from  any  scale  to  the  decimal  scale. 
1.    Express  34325  in  the  decimal  scale. 

3432 

5  EXPLANATION.  —  Since  each  higher  unit  is  equal  to  5  of 

T T  the  next  lower  order,  3  units  of  the  fourth  order  are  equal 
to  15  of  the  third,  and  adding  4,  the  number  of  the  third 

5  order  given,  we  obtain  19,  the  number  of  the  third  order. 

^  Proceeding  in  the  same  manner,  until  the  number  of 

^  units  of  the  first  order  is  obtained,  the  number  in  the 

5  decimal  scale  is  492. 
492 

Change  the  following  to  the  decimal  scale : 

2.  58679;  231234;   342546;  523647. 

3.  34326;  231*5U;  413245;  413426. 

4.  67358;  3819e12;  345147;  268^. 

543.  Arithmetical  processes  in  any  scale. 

The  processes  are  performed  in  the  same  manner  as  in  the  decimal 
scale.  The  student  must  simply  bear  in  mind  each  time  the  number 
of  units  of  each  order  required  to  make  one  of  the  next  higher  order. 

1.  Add  31235,  41245,  32435,  42335. 

2.  Add  52437,  62317,  56347,  35437. 

3.  Add  43849,  52769,  83469,  74369. 

4.  Subtract  345627  from  624567. 

5.  Subtract  413758  from  732458. 

6.  Multiply  34245  by  2346. 


PROOFS. 


FUNDAMENTAL  PROCESSES. 

544.  The  proofs  given  under  addition,  subtraction,  mul- 
tiplication, and  division  are  the  most  practical  and  reliable 
that  can  be  given.     A  briefer  method,  however,  has  been 
discovered,  which  may  be  employed  as  a  test  of  accuracy. 

545.  Method  by  casting  out  the  nines. 

It  has  been  discovered  that  when  the  number  of  9's  in  a 
number  is  found,  the  remainder  is  equal  to  the  sum  of  the 
digits  of  the  number,  or  the  sum  with  the  9's  omitted. 

Thus,  743  =  700  -T-  9  -f  40  -i-  9  +  3  ;  and  the  remainders  in  each 
instance  correspond  with  the  digits  which  express  the  number.  Hence 
the  sum  of  the  digits  7  +  4  +  3,  or  14,  or  (with  the  9  omitted)  5  is  the 
number  remaining  after  the  9's  have  been  found. 

546.  The  method  of  proof  by  casting  out  the  9's  is  based 
upon  the  presumption,  that  when  the  remainders  in  the 
results  agree  with  the  remainders  in  numbers,  from  which 
the  results  were  obtained,  the  work  is  correct. 

PROOF  OF  ADDITION. 

547.  1.   Prove  that  893  +  296  +  452  +  368  =  2009. 

893  2  EXPLANATION.  —  The  9's  in  the  first  addend  are  a 
296  8  certain  number  and  2  units  remaining ;  in  the  second,  a 
certain  number  and  8  units  ;  in  the  third,  a  certain  num- 
4oJ  Z  j.)er  an^  2  units ;  in  the  fourth,  a  certain  number  and  8 
368  8  units.  The  sum  of  the  units  remaining  is  20,  or  casting 
9  out  the  9's  from  tnat  sum  ft  is  2-  The  remainder  after 
£  casting  out  the  9's  in  the  sum  2009  is  also  2.  Hence, 
the  work  is  probably  correct. 
400 


FUNDAMENTAL  PROCESSES.  401 

It  should  be  borne  in  mind,  however,  that  this  is  not  an  accurate 
test  of  correctness,  for  the  same  excess  of  nines  will  be  obtained  in 
whatever  order  the  figures  are  arranged. 

PROOF  OF  SUBTRACTION. 

548.  Prove  that  18945  -  9326  =  9169.  • 

18945       0        EXPLANATION.  —  Casting  out  the  9's  from  the  minuend 
9326      2    there  is  °  for  a  remainder.     Casting  out  the  9's  from  the 

—  subtrahend  there  is  a  remainder  of  2,  2  subtracted  from 
*     a  unit  of  the  next  higher  order,  or  9,  leaves  a  remainder 

Casting  out  the  9's  from  the  remainder  there  is  also  a  remainder 
Hence  the  result  is  presumed  to  be  correct. 

PROOF  OF  MULTIPLICATION. 

549.  Prove  that  718  x  28  =  20104. 

718       7  EXPLANATION.  —  Casting  out  the  9's  from  the  multi- 

2g       -i  plicand  the  remainder  is  7.    Casting  out  the  9's  from  the 

—  multiplier,  the  remainder  is  1.    The  product  of  these 
^*       '  remainders  is  7,  and  it  is  the  same  as  the  remainder  after 

the  9's  have  been  cast  out  from  the  product.    Hence  the  work  is  proba- 
bly correct. 

PROOF  OF  DIVISION. 

550.  Prove  that  8232  -j-  21  =  392. 

365         EXPLANATION.  — Casting  the  9's  out  of  the  divisor 
21)  8232  (39 2,     an^  quotient,  the  remainders  are  3  and  5  respectively. 
Their  product  is  15,  from  which,  when  the  9's  are  cast  out,  the  remainder  , 
is  6.    This  number  corresponds  with  the  remainder  in  the  dividend' 
after  the  nines  have  been  cast  out  of  it ;  and  the  work  is  presumed  to  \ 
be  correct,  since  the  divisor  multiplied  by  the  quotient  is  equal  to  the 
dividend. 

643)5926431(9216  Eem.  543. 

EXPLANATION.  —  The  product  of  the  divisor  by  the  quotient,  plus 
the  remainder  is  equal  to  the  dividend.  Casting  the  9's  out  of  the 
divisor  the  remainder  is  4  ;  casting  them  out  of  the  quotient  the  re- 
mainder is  0.  The  product  of  4  and  0  is  0 ;  to  which  is  added  the 
excess  of  9's  in  the  remainder  which  is  3.  The  number  remaining 
after  casting  out  the  9's  from  the  dividend  is  3,  therefore  since  the 
results  agree,  the  work  is  presumed  to  be  correct. 

STAND.   AR.  —  26 


402  DIVISION  BY  FACTORS. 

DIVISION  BY  FACTORS. 
551.   1.   What  are  the  factors  of  32?    25?     64  ?     96  ? 

2.  If  a  number  is  divided  by  8,  by  what  must  the  quotient 
be  divided  that  the  number  may  be  divided  by  16  ? 

3.  If  a  number  is  divided  by  8  and  the  quotient  by  6, 
t  by  what  is  the  number  divided  ? 

4.  What  factors  may  be  used  to  divide  a  number  by  36  ? 

5.  What  factors  may  be  used  to  divide  a  number  by  48  ? 

6.  Divide  5683  by  32,  using  factors. 

5683  EXPLANATION.  —32  is  equal  to  4  x 

1420  3  2x4.     Dividing  5683  by  4  gives  a 

quotient  of  1420  fours  and  3  units 


710 


remaining. 


177  ...  2  Dividing  1420  fours  by  2  gives  a 

3  _1_  (2  x  8)  =  19   true  Rem.        quotient  of  710  eights.     Dividing  710 

17719.    Quotient  €^ktS   ^   4    Sives    a   <luotient   °f   177 

thirty-twos  and  2   eights   remainder. 

The  first  partial  remainder  is  3  units,  and  the  second,  2  eights,  or  16 ; 
hence,  the  entire  remainder  is  3  +  16,  or  19,  and  the  quotient  is 


RULE.  —  Divide  the  dividend  by  one  factor  of  the  divisor, 
the  quotient  thus  obtained  by  another  factor,  and  so  continue 
until  all  the  factors  have  been  used  successively  as  divisors. 

If  there  are  remainders,  multiply  each  remainder  by  all  the 
preceding  divisors  except  the  one  that  produced  it.  The  sum 
of  these  products  will  be  the  true  remainder. 

Divide,  using  factors : 

7.  1704  by  24.       11.  1288  by  56.  15.  3275  by  56. 

8.  4725  by  15.        12.  3528  by  72.  16.  3276  by  27. 

9.  5740  by  28.        13.  3824  by  32.  17.  4104  by  45. 
10.    1428  by  42.       14.  2184  by  49.  18.  7304  by  24. 


MEASUEEMENT    OF    SOLIDS. 


552.  A  surface  such  that  a  straight  line  joining  any  two 
points  of  it  lies  wholly  in  the  surface  is  a  Plane  Surface. 

553.  A  surface  no  part  of  which  is  a  plane  surface  is  a 
Curved  Surface. 

554.  A  plane  figure  bounded  by  straight  lines  is  a  Poly- 
gon. 

555.  The  length  of  the  lines  that  bound  a  figure  is  its 
Perimeter. 

556.  Anything  that  has  length,  breadth,  and  thickness  is 
a  Solid  or  Body. 

The  plane  surfaces  or  planes  which  bound  a  solid  are  called  its 
/aces,  and  their  intersections,  its  edges. 


Parallclopipcdou. 


Cylinder. 


557.   A  solid  whose  two  ends  are  equal  polygons,  parallel 
to  each  other,  and  whose  sides  are  parallelograms,  is  a  Prism. 

Prisms,  from  the  form  of  their  bases,  are  named  triangular, 
quadrangular,  pentagonal,  etc. 

403 


404 


MEASUREMENT  OF  SOLIDS. 


558.  A   regular  solid  bounded  by  a  uniformly  curved 
surface,  and  having  for  its  ends  two  equal  circles,  parallel  to 
each,  other,  is  a  Cylinder. 

The  face  of  any  section  of  a  cylinder  parallel  to  the  base  is  a  circle 
equal  to  the  base. 

559.  A  solid  whose  base  is  a  polygon  and  whose  faces 
are  triangles,  meeting  at  a  point  called  the  vertex,  is  a 
Pyramid. 

560.  A  solid,  whose  base  is  a  circle  and  whose  surface 
tapers  uniformly  to  a  point  called  the  vertex,  is  a  Cone. 


Pyramid.  Cone.  Frustum  of  Cone.    Frustum  of  Pyramid. 

561.  The  portion  remaining  after  the  top  has  been  cut 
off  from  a  pyramid  or  cone  by  a  plane  parallel  to  the  base, 
is  a  Frustum  of  a  pyramid  or  cone. 

562.  A  solid,  every  point  of  whose  surface  is  equally 
distant  from  a  point  within,  called  the  center,  is  a  Sphere. 

563.  A  straight  line  passing  through  the 
center,  and  terminating  in  the  surface  of  a 
sphere  at  both  ends,  is  its  Diameter. 

564.  One  half  the  diameter,  or  the  dis- 
tance  from  the  center  to  the  surface  of  a 

sphere,  is  its  Radius.  Sphere. 

565.  The  greatest  distance  around  a  sphere  is  its  Circum- 
ference. 

566.  The  perpendicular  distance  from  the  highest  point 
of  a  solid  to  the  plane  of  the  base  is  its  Altitude. 


SURFACE  OF   SOLIDS.  405 


SURFACE    OP   SOLIDS. 

567.  All  the  surface  of  a  solid  except  its  base  or  bases 
is  called  the  Lateral  Surface. 

The  entire  surface  includes  the  area  of  the  bases  also. 

WRITTEN    EXERCISES. 

568.  To  find  the  lateral  surface  of  a  prism  or  cylinder. 

It  .is  evident  that  if  a  prism  or  cylinder  were  1  inch 
high,  its  lateral  surface  would  contain  as  many  square 
units  of  surface  as  there  were  units  in  the  perimeter  of 
the  base ;  and  if  it  were  2  inches,  3  inches,  or  4  inches 
high,  the  lateral  surface  would  contain  2,  3,  or  4  times 
the  number  of  units  in  the  perimeter  of  the  base.  Hence 
the  following  rule : 

KULE.  —  Multiply  the  perimeter  of  the  base  by  the  altitude. 

1.  What  is  the  lateral  surface  of  a  cylinder  whose  diam- 
eter is  2  feet,  and  whose  length  is  5  feet? 

2.  What  is  the  lateral  surface  of  a  quadrangular  prism 
whose  sides  are  each  2£  feet,  and  whose  height  is  4  feet  ? 

3.  What  is  the  lateral  surface  of  a  triangular  prism  whose 
sides  are  each  6  feet,  and  whose  altitude  is  8  feet  ? 

4.  What  is  the  entire  surface  of  a  cylinder  which  is  5 
feet  in  length,  and  whose  base  is  2  feet  in  diameter  ? 

569.  To  find  the  lateral  surface  of  a  pyramid  or  cone. 

It  is  evident  that  the  lateral  surface  of  any  pyra- 
mid is  composed  of  triangles,  and  the  lateral  surface 
of  a  eone  may  also  be  assumed  to  be  made  up  of  an 
infinite  number  of  triangles.  The  bases  of  these 
triangles  form  the  perimeter  of  the  solid,  and  their 
height  is  the  slant  height  of  the  solid.  Hence  the 
following  rule : 

KULE.  —  Multiply  the  perimeter  of  the  base  by  one  half  the 
slant  height. 


406  MEASUREMENT  OF  SOLIDS. 

1.  What  is  the  lateral  surface  of  a  quadrangular  pyra- 
mid whose  base  is  15  feet  square,  and  whose  slant  height 
is  18  feet  ? 

2.  What  is  the  lateral  surface  of  a  cone  whose  diameter 
at  the  base  is  12  feet,  and  whose  slant  height  is  20  feet  ? 

3.  What  is  the  lateral  surface  of  a  cone  whose  base  is 
20  feet  in  diameter,  and  whose  slant  height  is  20  feet  ? 

4.  What  is  the  cost  of  painting  a  church  steeple,  the  base 
of  which  is  an  octagon  6  feet  on  each  side,  and  whose  slant 
height  is  80  feet,  at  $  .30  per  square  yard  ? 

5.  How  many  feet  of  lateral  surface  are  there  on  a  cone, 
the  base  diameter  of  which  is  6  feet,  and  whose  slant  height 
is  9£  feet  ? 

6.  How  many  feet  of  lateral  surface  are  there  on  a  pyra- 
mid whose  base  is  10  feet  square,  and  whose  slant  height 
is  20  feet  ? 

7 .  How  many  feet  of  lateral  surface  are  there  on  a  cone 
whose  base  is  8  feet  in  diameter,  and  whose  slant  height  is 
6  feet  ? 

8.  What  is  the  lateral  surface  of  a  cone  whose  base  is 
10  feet  in  diameter,  and  whose  slant  height  is  10  feet  ? 

570.   To  find  the  lateral  surface  of  a  frustum  of  a  pyramid 
or  cone. 

It  is  evident  that  the  lateral  surface  of  a  frustum 
of  a  pyramid  is  composed  of  trapezoids,  the  sum  of 
whose  parallel  sides  forms  the  perimeter  of  the  bases, 
and  whose  altitude  is  the  slant  height  of  the  frustum  ; 
and  the  lateral  surface  of  a  cone  may  be  assumed  to 
be  made  of  an  infinite  number  of  trapezoids.  Hence 
the  following  rule : 

RULE.  —  Multiply  half  the  sum  of  the  perimeter  of  the  two 
bases  by  the  slant  height. 


SURFACE  OF  SOLIDS.  407 

1.  How  many  feet  of  lateral  surface  are  there  in  the 
frustum  of  a  cone  whose  slant  height  is  8  feet,  the  diameter 
of  whose  lower  base  is  12  feet  and  of  the  upper  base  8  feet  ? 

2.  What  is  the  lateral  surface  of  the  frustum  of  a  pyra- 
mid, the  slant  height  of  which  is  25   feet,  whose   lower 
base  is  40  feet  square,  and  whose  upper  base  is  20  feet 
square  ? 

3.  What  did  it  cost,  at  $.15  per  square  yard,  to  paint  the 
lateral  surface  of  a  vat  which  was  10  feet  in  diameter  at 
the  bottom  and  8  feet  at  the  top,  the  slant  height  of  which 
was  12  feet  ? 

4.  What  is  the  lateral  surface  of  a  vat,  the  base  of  which 
is  9  feet  square,  whose  top  is  8  feet  square,  and  whose  slant 
height  is  10  feet  ? 

5.  What  will  be  the  cost  of  Covering  the  outside  surface, 
including  the  bottom  of  such  a  vat  with  sheet  metal  at  8 
cents  per  square  foot  ? 

571.   To  find  the  convex  surface  of  a  sphere. 

The  convex  surface  of  a  sphere  is  computed,  according  to  geomet- 
rical principles,  by  the  following  rules : 

RULE.  —  1.  Multiply  the  diameter  by  the  circumference. 
2.    Multiply  the  square  of  the  diameter  by  3.1416. 

1.  What  is  the  convex  surface  of  a  sphere  whose  diam- 
eter is  15  inches  ? 

2.  What  is  the  convex  surface  of  a  spherical  cannon-ball 
8  inches  in  diameter  ? 

3.  What  is  the  convex  surface  of  a  base-ball  whose  cir- 
cumference is  9^  inches  ? 

4.  What  is  the  convex  surface  of  a  sphere  whose  cir- 
cumference is  12  feet  ? 


408  MEASUREMENT  OF   SOLIDS. 


VOLUME  OP  SOLIDS. 

572.   The  Volume  of  any  body  is  the  number  of  solid  units 
it  contains. 

WRITTEN   EXERCISES. 
573.   To  find  the  volume  of  a  prism  or  cylinder. 

It  is  evident  that  if  a  prism  or  cylinder  were  J  inch 
high,  it  would  contain  as  many  cubic  inches  as  there 
were  square  inches  in  the  area  of  the  base ;  and  if  it 
were  2  inches,  3  inches,  or  4  inches  high,  the  volume 
would  be  2  or  3  or  4  times  as  much.  Hence  the  fol- 
lowing rule : 

EULE. — Multiply  the  area  of  the  base  by  the  altitude. 

1.  What  are  the  solid  contents  of  a  prism  whose  base 
is  12  inches  square  and  whose  height  is  2  feet  ? 

2.  What  is  the  volume  of  a  cylinder  whose  diameter  is 
11  feet  and  whose  length  is  4  feet  ? 

3.  What  will  be  the  cost  of  a  piece  of  timber  20  feet 
long,  18  inches  wide,  and  12  inches  thick  at  $  .30  per  cubic 
foot? 

4.  What  was  the  capacity  in  bushels  of  a  square  bin, 
the  base  of  which  was  8  feet  square,  and  the  height  of 
which  was  9  feet  on  the  inside  ? 

5.  How  many  gallons  of  water  will  a  vat  in  the  form  of 
a  cylinder  hold,  whose  inside  dimensions  are  —  base  8  feet 
in  diameter,  height  7  feet. 

6.  How  much  would  the  wheat  be  worth  at  $  1.85  per 
bushel,  which  would  just  fill  a  bin  the  base  of  which  is  15 
feet  square,  and  the  height  of  which  is  12  feet  ? 


VOLUME   OF   SOLIDS.  409 

574.  To  find  the  volume  of  a  pyramid  or  cone. 

It  can  be  shown  by  geometry  that  a  pyramid  or  cone  is  one  third 
of  a  prism  or  cylinder  of  the  same  base  and  altitude.  Hence  the  fol- 
lowing rule : 

RULE.  —  Multiply  the  area  of  the  base  by  one  third  of  the 
altitude. 

1.  What  are  the  solid  contents  of  a  cone,  the  diameter  of 
whose  base  is  6  feet,  and  whose  altitude  is  9  feet  ? 

2.  What  are  the  solid  contents  of  a  pyramid  whose  base 
is  30  feet  square  and  whose  altitude  is  60  feet  ? 

3.  If  a  cubic  foot  of  granite  weighs  165  pounds,  what  is 
the  weight  of  a  granite  cone  the  diameter  of  whose  base 
is  6  feet,  and  whose  altitude  is  8  feet  ? 

4.  What  is  the  weight  of  a  marble  pyramid  whose  base 
is  4  feet  square  and  whose  altitude  is  8  feet,  if  a  cubic  foot 
of  marble  weighs  171  pounds  ? 

575.  To  find  the  volume  of  a  frustum  of  a  pyramid  or  cone. 

It  can  be  shown  by  geometry  that  the  frustum  of  a  pyramid  or  cone 
is  equal  to  three  pyramids  or  cones,  having  for  their  bases,  respectively, 
the  upper  base  of  the  frustum,  its  lower  base,  and  a  mean  proportional 
between  the  two  bases.  Hence  the  following  rule : 

RULE.  —  To  the  sum  of  the  areas  of  the  two  ends  add  the 
square  root  of  the  product  of  these  areas,  and  multiply  the 
result  by  one  third  of  the  altitude. 

1.  What  is  the  volume  of  a  frustum  of  a  pyramid  the 
lower  base  of  which  is  20  feet  square,  the  upper  base  10 
feet  square,  and  the  altitude  20  feet  ? 

2.  What  are  the  solid  contents  of  the  frustum  of  a  cone 
whose  upper  base  is  5  feet  in  diameter,  whose  lower  base  is 
8  feet  in  diameter,  and  whose  altitude  is  7  feet  ? 


410  MEASUREMENT  OF  SOLIDS. 

3.  A  tree  was  3  feet  in  diameter  at  the  butt,  and  its 
diameter  at  a  height  of  40  feet  was  1  foot.     What  were  the 
cubical  contents  of  that  portion  of  the  tree  ? 

4.  A  vat  whose  inside  measurements  were  as  follows  — 
diameter  of  the  bottom  12  feet,  diameter  of  the  top  10  feet, 
height  9  feet  —  was  filled  with  water.     How  many  gallons 
did  it  contain  ? 

576.   To  find  the  volume  or  contents  of  a  sphere. 

A  sphere  may  be  regarded  as  composed  of  pyra- 
mids whose  bases  form  the  surface  of  the  sphere, 
and  whose  altitude  is  the  radius  of  the  sphere. 
Hence  the  following  rule : 

RULE.  —  1.  Multiply  the  convex  surface 
by  one  third  of  the  radius;  or, 

2.   Multiply  the  cube  of  the  diameter  by  .5236. 

1.  The  diameter  of  a  sphere  is  5  feet.     How  many  cubic 
feet  does  it  contain  ? 

2.  Find  the  contents  of  a  sphere  whose  diameter  is  8  feet. 

3.  The  circumference  of  a  sphere  is  9.4248.     What  are 
its  cubical  contents  ? 

4.  A  cubic  foot  of  cast-iron  weighs  about  450  pounds. 
What  is  the  weight  of  a  cannon-ball  whose  diameter  is  18 
inches  ? 

5.  What  are  the  cubical  contents  of  a  spherical  vessel 
the  diameter  of  which  is  2A-  feet  ? 

6.  How  many  cubic  feet  are  there  in  a  spherical  body 
whose  diameter  is  25  feet  ? 

7.  If  a  cubic  inch  of  water  weighs  252.96  gr.,  and  iron  is 
7.21  times  as  heavy  as  water,  what  will  be  the  weight  of  a 
six-inch  cannon  ball  ? 


METRIC    SYSTEM    OF   WEIGHTS   AND 
MEASURES. 


577.  The  Metric  System  of  weights  and  measures  is  used 
by  most  of  the  civilized  nations  of  the  world  except  the 
United  States  and  Great  Britain  and  some  of  her  colonies. 
It  has  also  been  legalized  by  the  United  States  government. 

578.  The  unit  of  length  is  the  meter,  and  from  it  the  other 
units,  viz. :  surface,  volume,  capacity,  and  weight  are  derived. 

1.  The  length  of  the  meter  was  intended  to  be  one  ten-millionth  of 
the  distance  from  the  equator  to  the  poles,  but  subsequent  calculations 
have  shown  it  to  be  a  very  little  less  than  that. 

2.  The  system  derives  its  name  from  the  meter,  because  all  the  units 
of  measure  in  the  system  are  derived  from  it. 

3.  The  system  has  a  decimal  notation. 

579.  From  the  standard  units  other  denominations  are 
formed  by  the  use  of  prefixes  derived  from  the  Latin  and 
the  Greek. 

Deci   means  10th.  Deka    means  10. 

Centi  means  100th.  Hekto  means  100. 

Milli  means  1000th.  Kilo     means  1000. 

Myria  means  10000. 
Thus,  decimeter,  -fa  of  a  meter ;  dekameter,  10  meters. 

580.  The  following  tables  give  all  the  denominations,  but 
many  of  them  are  not  in  common  use.     Those  usually  em- 
ployed in  business  or  science  are  indicated  by  bold-faced  type. 

Abbreviations  beginning  with  a  small  letter  indicate  a 
fractional  part  of  the  standard  unit ;  those  beginning  with 
a  capital  denote  a  multiple  of  the  unit. 

411 


412  METRIC   SYSTEM. 

METRIC   TABLES. 
MEASURES  OF  LENGTH. 

581.  The  unit  of  length  is  the  meter. 

TABLE. 

10  Millimeters  (mm)  =  1  Centimeter  (cm)  =  .3937079  in. 

10  Centimeters  =  1  Decimeter  (*"»)  =  3.937079  in. 

10  Decimeters  =  1  Meter  (m)  =  39.37079  in. 

10  Meters  =  1  Dekameter  (Dm)  =  32.80899  ft. 

10  Dekameters  =  1  Hektometer  (Hm)  =  19.92781  rd. 

10  Hektometers          =  1  Kilometer  (Km)  =  .621382  mi. 

10  Kilometers  =  1  Myriameter(Mra)  =  6.21382  mi. 

MEASURES  OF  SURFACE. 

582.  The  units  of  surface  are  squares  whose  dimensions 
are  the  corresponding  linear  units;    hence  it  requires  10 
times  10,  or  100,  of  a  given  denomination  to  make  one  of  the 
next  higher. 

TABLE. 

100  Sq.  Millimeters    =  1  Sq.  Centimeter  =  .155+  sq.  in. 

100  Sq.  Centimeters  =  1  Sq.  Decimeter  =  15.5+  sq.  in. 

100  Sq.  Decimeters     =  1  Sq.  Meter  =  1.196+  sq.  yd. 

100  Sq.  Meters  =  1  Sq.  Dekameter  =  119.6034  sq.  yd. 

100  Sq.  Dekameters   =  1  Sq.  Hektometer  =  2.47114  A. 

100  Sq.  Hektometers  =  1  Sq.  Kilometer  =  247.114  A.  =  .3861  sq.  mi. 

1.  In  measuring  small  areas  the  unit  is  the  sq.  meter. 

2.  In  measuring  land,  the  square  meter  is  called  a  centare  (ca),  the 
square  dekameter  an  are  (a)  and  the  square  hektometer  a  hektare  (Ha). 

MEASURES  OF  VOLUME. 

583.  The  units  of  volume  are  cubes  whose  dimensions 
are  the  corresponding  linear  units ;  hence  it  requires  1000 
of  a  given  denomination  to  make  one  of  the  next  higher. 


MEASURES  OF  CAPACITY  AND  WEIGHT.      413 

TABLE. 

1000  Cu.  Millimeters  (cumm)_  j  cu.  Centimeter  (•»«»). 
1000  Cu.  Centimeters  =  1  Cu.  Decimeter  (cudm). 

1000  Cu.  Decimeters  =  1  Cu.  Meter  (cum). 

1.  In  measuring  volumes  the  principal  unit  is  the  cubic  meter. 

2.  In  measuring  wood,  the  cubic  meter  is  called  a  stere,  TV  of  a 
cubic  meter  a  decistere,  etc. 

3.  A  cubic  meter  is  equal  to  1.308  cubic  yards. 

MEASURES   OF  CAPACITY 

584.  The  unit  of  capacity  in  both  liquid  and  dry  measure 
is  the  liter,  and  it  contains  a  volume  equal  to  a  cube  whose 
edge  is  a  decimeter. 

TABLE. 

LIQUID.  DBT. 

10  Milliliters  (ml)  =  1  Centiliter  (cl)  =  .6102  cu.  in. 
10  Centiliters        =  1  Deciliter  (<«)    =  .845  gi. 
10  Deciliters         =  1  Liter  Q  =  1.0567  qt.  .908  qt. 

10  Liters  =  1  Dekaliter  (Di)  =  2.6417  gal.         1.135  pk. 

10  Dekaliters        =  1  Hektoliter  (Hi)  =  26.417  gal.      2.8375  bu. 
10  Hektoliters      =  1  Kiloliter  (K1)     =  264.17  gal. 

MEASURES   OF  WEIGHT. 

585.  The  unit  of  weight  is  the  gram,  and  its  weight  is 
the  weight  of  a  cubic  centimeter  of  distilled  water  at  its 
greatest  density. 

TABLE. 

AVOIRDUPOIS. 

10  Milligrams  (m«)  =  1  Centigram  («*)  =  .15432  +  gr. 
10  Centigrams  =  1  Decigram  (ds)  =  1. 54324 +  gr. 
10  Decigrams  =  1  Gram  («)  =  15.43248  +  gr. 

10  Grams  =  1  Dekagram  (D*)    =         .35273  +  oz. 

10  Dekagrams  =  1  Hektogram  (HK)  =  3.52739  +  oz. 
10  Hektograins  =  1  Kilogram  (K<?)  =  2.20462  +  Ib. 
10  Kilograms  =  1  Myriagram  (Mfc)  =  22.04621  +  Ib. 

10  Myriagrams  =  1  Quintal  (Q)  =  220.46212  +  Ib. 
10  Quintals  =  1  Tonneau  (T)  =  2204.62125  +  Ib. 


414  METRIC   SYSTEM. 


REDUCTION. 

586.  A  denominate  number  in  the  metric  system  can  be 
reduced  to  higher  or  lower  denominations  by  the  removal 
of  the  decimal  point. 

1.  In  measures  of  volume,  since  it  requires  1000  of  each 
denomination  to  make  one  of  the  next  higher,,  the  decimal  point 
must  be  removed  three  places. 

2.  In  measures  of  surfaces,  since  it  requires  100  of  each 
denomination  to  make  one  of  the  next  higher,  the  decimal  point 
must  be  removed  two  places. 

3.  In  other  measures  remove  the  decimal  point  one  place. 

EXERCISES. 

587.  1.   Eeduce  15675cm  to  kilometers. 

2.  Eeduce  75608qm  to  ares. 

3.  Write  6734cl  as  liters  ;  as  hektoliters. 

4.  Write  43628mg  as  grams ;  as  kilograms. 

5.  Write  .75CU  m  as  liters ;  as  hektoliters. 

6.  Write  876.378qcm  as  square  meters;  as  ares. 

7.  What  is  the  weight  of  230.5cucm  of  water  ? 
A  cubic  centimeter  of  water  weighs  1  gram. 

8.  What  is  the  weight  in  dekagrams  of  .045cum  of  water? 

9.  What  is  the  weight  in  kilograms  of  13H1  of  water? 

10.  How  much  will  a  cubic  meter  of  water  weigh  in 
kilograms  ?    Express  the  same  quantity  of  water  in  liters. 

11.  What  is  the  amount  of  65750.751  of  water  in  cubic 
meters  ?     What  is  its  weight  in  kilograms  ? 

A  milliliter  of  water  weighs  1  gram. 

12.  What  will  60l  of  mercury  weigh,  mercury  being  13.5 
times  as  heavy  as  water  ? 

The  weight  of  any  substance  compared  with  the  weight  of  an  equal 
bulk  of  water  is  called  its  specific  gravity.  The  sp.  gr.  of  mercury  is 
13.5. 


REDUCTION.  415 

13.  Find  the  weight  in  grams  of  a  cubic   decimeter  of 
iron  (sp.  gr.  7.21). 

14.  What  is  the  weight  in  kilograms  of  a  cubic  meter  of 
ice  (sp.  gr.  0.92)? 

15.  Which  is  the  cheaper  and  how  much,  to  buy  cloth  at 
$  3  per  meter,  or  at  $2.90  per  yard  ? 

16.  How  many  ares  are  there  in  a  rectangular  field  62m 
long  and  43.6m  wide  ? 

17.  Find  in  square  meters  the  area  of  the  floor  of  a  room 
5.3  meters  long  and  4.2  meters  wide. 

18.  How  many  liters  of  water  are  there  in  a  tank  2.6m 
long,  2m  wide,  and  6dm  deep  ? 

19.  How  long  must  a  pile  of   wood  be,  to  contain   12 
steres,  if  it  is  3.5  meters  high  and  3.8  meters  wide  ? 

20.  A  barrel  of  flour  contains  196  pounds.     How  many 
kilograms  or  kilos  does  it  weigh  ? 

21.  A  bin  is  3  meters  square  and  2.5  meters  high.     How 
many  hektoliters  of  wheat  will  it  hold  ? 

22.  What  weight  of  mercury  will  a  vessel  contain  whose 
capacity  is  25cucm? 

23.  A  tank  is  4m  long,  36dm  wide,  76cm  deep.     How  many 
liters  of  water  will  it  hold  ? 

24.  A  vat  is  6.4m  long,  3m  wide,  and  2.8m  deep.     How 
long  will  it  take  a  water-pipe  to  fill  the  vat,  if  2.9D1  flow 
into  it  per  minute  ? 

25.  A  platform  bears  a  weight  of  60  pounds  per  square 
foot.     What  is  the  weight  in  kilograms  per  square  meter  ? 

26.  A  man  bought  360  bu.  of  wheat  at  $  .95  a  bushel, 
;and  sold  it  at  $  2.95  a  hektoliter.     How  much  did  he  gain  ? 

27.  The  dimensions  of  a  box  are  3.5m,  1.8m,  and  0.8m. 
What  are  the  contents  in  cubic  yards  ? 

28.  A  room  is  5.2  meters  long,  4.5  meters  wide,  and  3.2 
meters  high.     What  will  be  the  cost  of  plastering  it  at  35 
•cents  per  square  meter  ? 


416  METRIC   SYSTEM. 

29.  How  much  will  a  merchant  receive  for  3.68m  of  wine, 
if  he  sells  it  at  $  2.50  per  gallon  ? 

30.  What  is   the   weight   in  kilograms   of  2583  cucm  of 
water  ? 

31.  How  much  will  4  tons  of  coal   cost   at  $.75  per 
quintal  ? 

32.  Reduce  20  Ib.  8  oz.  Avoirdupois  to  hektograms. 

33.  How  many  revolutions  will  be  made  by  a  wheel  9  ft. 
in  circumference  in  passing  over  a  distance  of  one  kilometer  ? 

34.  Which  is  the  larger   and  how  much,  a  cask  which 
holds  21D1  or  one  which  holds  43  gallons  ? 

35 .  Change  10  hektograms  to  pounds  Avoirdupois  weight ; 
Troy  weight. 

36.  Find  the  weight  in  grams  of  a  gallon  of  water. 

37.  What  price  per  pound  is  equivalent  to  $2.20  per 
kilogram  ? 

38.  A  square  foot  is  what  part  of  a  square  meter  ? 

39.  How  many  centares  are  there  in  a  garden  plot  10  feet 
square  ? 

40.  The  specific  gravity  of  granite  is  2.9.     What  is  the 
weight  of  a  block  50cm  long,  25cm  wide,  12cm  thick,  in  kilo- 
grams ?   in  pounds  ? 

41.  What  is  the  weight  in  kilograms  of  a  cubic  foot  of 
water  ? 

42.  How  many  meters  are  there  in  2  mi.  40  rd.  12  ft.  ? 

43.  How  many  gallons  are  there  in  24  dekaliters  ? 

44.  A  man  bought  50Kg  of  sugar  for  $  5.51.     How  much 
did  he  pay  per  pound  ? 

45.  How  many  gallons  will  a  cistern  contain  which  is  3m 
square  and  2ra  deep  ? 

46.  What  price  per  bushel  is   equivalent  to  $6.60  per 
hektoliter  ? 

47.  If  the  specific  gravity  of  copper  is  8.8,  what  is  the 
weight  in  hektograms  of  a  cubic  decimeter  of  the  metal  ? 


REDUCTION.  417 

48.  If  a  bushel  of  oats  weighs  32  pounds,  what  is  the 
weight  in  kilograms  of  40  bu.  ? 

49.  If  a  body  weighs  7.35Kg  in  air  and  4.41Kg  in  water, 
what  is  its  specific  gravity  ? 

50.  What  is  the  weight  in  dekagrams  of  44  cubic  deci- 
meters of  zinc,  if  the  specific  gravity  is  6.86  ? 

51.  A  cask  of  olive  oil  containing  2H1  cost  $  36.32.    What 
was  paid  per  quart  for  the  oil  ? 

52.  If  a  silver   dollar  weighs   4121  grains,  how  many 
grams  does  it  weigh  ? 

53.  If  a  quire  of  paper  is  .588cm  thick,  what  is  the  thick- 
ness in  millimeters  of  a  single  sheet  ? 

54.  What  is  the  value  of  20  qt.  of  sulphuric  acid  at  2J 
cents  a  pound,  if  the  specific  gravity  is  1.841  ? 

55.  A  rectangular  vessel  is  5m  long,  9dm  wide  and  3m  deep. 
Find  its  capacity  in  cubic  meters.     What  is  the  weight  of 
distilled  water  at  its  greatest  density  that  it  will  hold. 

56.  A  liter  is  .264  of  a  gallon.     How  many  grams  will  a 
gill  of  water  make  ? 

57.  A  rectangular  cistern  is  known  to  hold  25H1.     If  its 
length  is  2m,  its  breadth  1.5m,  what  is  its  depth  ? 

58.  An  importer  bought  silk  at  $  1.15  per  meter  and  sold 
it  at  a  profit  of  20%  per  yard.     How  much  did  he  get  for 
it  per  yard  ? 

59.  Mercury  is  131  times  as  heavy  as  water,  or  its  specific 
gravity   is   13.5.      How  many   pounds  will   a  vessel   hold 
whose  capacity  is  35CU  cm  ? 

60.  A  man  paid  800  francs  for  100  liters  of  wine.     What 
price  did  he  pay  for  it  per  gallon  ? 

61.  A  cubical  block  of  silver  whose  dimensions  are  2dm 
was  sold  at  20^  per  dekagram.     If  the  specific  gravity  of 
silver  is  10.5,  for  how  much  did  it  sell  ? 

62.  If  alcohol  is  .8  as  heavy  as  water,  how  many  pounds 
Avoirdupois  will  1250cucm  of  alcohol  weigh? 


TABLES  OF  DENOMINATE  NUMBERS. 


MEASURES  OP  EXTENSION. 

588.  Measures  of  Extension  are  used  in  measuring  lengths, 
distances,  surfaces,  and  solids. 

589.  LINEAR  MEASURE. 

TABLE. 

12  Inches  (in.)         =  1  Foot      .     .     .     .  ft. 

3  Feet  =  1  Yard      ....  yd. 

5|  Yards  or  IQ\  ft.  =  1  Rod rd. 

320  Rods  =  1  Mile       ....  mi. 

mi.      rd.          yd.  ft.  in. 

1  =  320  =  1760  =  5280  =  63360. 
Scale.—  320,  5£,  3,  12. 

The  following  are  also  used : 

3  Barleycorns  =  1  Inch.    Used  by  shoemakers. 

4  Inches  =1  Hand.   Used  to  measure  the  height  of  horses. 
6  Feet  =  1  Fathom.     Used  to  measure  depths  at  sea. 

f  ^eet  =  \  ^l6'  }     Used  in  pacing  distances. 

5  Paces  =  1  Rod.  / 

8  Furlongs  =  1  Mile. 

1.15  Statute  Miles        =  1  Geographical,  or  Nautical  Mile. 

3  Geographical  Miles  =  1  League. 

60  Geographic  Miles  1  _  i  T)          /of  Latitude  on  a  Meridian,  or 
69.16  Statute  Miles  /  ~  1  of  Longitude  on  the  Equator. 

1.  The  length  of  a  degree  of  latitude  varies.    69.16  miles  is  the 
average  length,  and  is  that  adopted  by  the  United  States  Coast  Survey. 

2.  The  standard  unit  of  length  is  identical  with  the  imperial  yard 
of  Great  Britain. 

3.  The  standard  yard,  under  William  IV.,  was  declared  to  be  fixed 
by  dividing  a  pendulum  which  vibrates  seconds  in  a  vacuum,  at  the 
level  of  the  sea,  at  62°  Fahrenheit,  in  the  latitude  of  London,  into 
391,393  equal  parts,  and  taking  360,000  of  these  parts  for  the  yard. 

The  following  denominations  also  occur :    The  span  =  9  inches ; 
1  common  cubit  (the  distance  from  the  elbow  to  the  end  of  the  middle 
finger)  =  18  inches ;  1  sacred  cubit  =  21.888  inches. 
418 


LINEAR  MEASURES.  419 

590.  SQUARE  MEASURE. 

TABLE. 

144  Square  Inches  (sq.  in.)  =  1  Square  Foot     .     .     .     sq.  ft. 
9  Square  Feet  =  1  Square  Yard     .     .     .     sq.  yd. 

30£  Square  Yards      .  =  1  Square  Rod       .     .     .    sq.  rd. 

160  Square  Kods  =  1  Acre A. 

640  Acres  =  1  Square  Mile      .     .     .     sq.  mi. 

sq.  mi.    A.         sq.  rd.  sq.  yd.  sq.  ft.  sq.  in. 

1  =  640  =  102400  =  3097600  =  27878400  =  4014489600. 
.  Scale.  —  640,  160,  30£,  9,  144. 

1.  Th'e  term  perch  or  pole  is  sometimes  used  instead  of  rod.    The 
rood,  40  perches,  is  found  in  old  title  deeds  and  surveys. 

2.  Plastering,  ceiling,  etc.,  are  commonly  estimated  by  the  square 
yard;  paving,  glazing,  and  stone-cutting,  by  the  square  foot;  roof- 
ing,  flooring,  and  slating  by  the  square  of  100  feet. 

591.  CUBIC   MEASURE. 

TABLE. 

1728  Cubic  Inches  (cu.  in.)  =  1  Cubic  Foot    .    .    .  ca.  ft. 

27  Cubic  Feet  =  1  Cubic  Yard    .    .    .  cu.  yd. 

128  Cubic  Feet  =  1  Cord C. 

cu.  yd.  cu.  ft.  cu.  in. 
1  =  27  =  46656. 
Scale.— 27,  1728. 

1.  A  cord  of  wood  or  stone  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft. 
high. 

2.  A  perch  of  stone  or  masonry  is  16£  ft.  long,  1£  ft.  thick,  and 
1  ft.  high,  and  contains  24|  cu.  ft. 

3.  A  cubic  yard  of  earth  is  considered  a  load. 

4.  Brick- work  is  commonly  estimated  by  the  thousand  bricks. 

6.  Brick-layers,  masons,  and  joiners  commonly  make  a  deduction 
of  one  half  the  space  occupied  by  windows  and  doors  in  the  walls  of 
buildings. 

6.  In  computing  the  contents  of  walls,  masons  and  brick-layers 
multiply  the  entire  distance  around  on  the  outside  of  the  wall  by  the 
height  and  thickness.    The  corners  are  thus  measured  twice. 

7.  A  cubic  foot  of  distilled  water  at  the  maximum  density,  at  the 
level  of  the  sea,  and  the  barometer  at  30  inches,  weighs  62£  Ib.  or 
1000  oz.  Avoirdupois. 


420          TABLES   OF   DENOMINATE  NUMBERS. 

592.  SURVEYORS'  LINEAR  MEASURE. 

TABLE. 

7.92  Inches  =  1  Link      ...    1. 

25  Links  =  1  Rod  ....    rd. 

4  Rods  or  100  Links  =  1  Chain    .     .     .    ch. 
80  Chains  =  1  Mile      ...    mi. 

mi.     ch.        rd.  1.  in. 

1  =  80  =  320  =  8000  =  63360. 
Scale.  —  80,  4,25,  7.92. 

1.  The  Linear  Unit  commonly  employed  by  surveyors  is  Gunter's 
Chain,  which  is  4  rods  or  66  feet. 

2.  An  Engineers'  Chain,  used  by  civil  engineers,  is  100  feet  long, 
and  consists  of  100  links. 

593.  SURVEYORS'  SQUARE  MEASURE. 

TABLE. 


625  Square  Links  =  1  sq.  rd. 


10  Square  Chains  =  1  acre. 
640  Acres  =  1  sq.  mi. 


16  Square  Rods   =  1  sq.  chain. 

eq.  mi.    A.        sq.  ch.        sq.  rd.  sq.  1. 

1  =  640  =  6400  =  102400  =  64000000. 
Scale.  —  640,  10,  16,  625. 

In  some  parts  of  the  country  a  Township  contains  36  square  miles, 
or  is  6  miles  square.    A  square  mile  of  land  is  also  called  a  section. 


1.  In  Texas,  New  Mexico,  and  other  Spanish  sections  of  the  United 
States,  the  Spanish  land  measures  are  still  in  use.    The  unit  of  length 
is  the  vara,  equal  in  Texas  to  33£  inches,  in  California  to  33  inches, 
and  in  Mexico  to  32.9927  inches.    Counting  33£  inches  to  the  vara,  108 
varas  =  100  yards,  and  1900.8  varas  =  1  mile. 

2.  Land  is  measured  in  square  varas,  labors,  and  square  leagues. 

1000000  Square  Varas  =  1  Labor  =  177.136  Acres. 

25  Labors  =  1  Square  League  =  4428.4  Acres. 

1  Acre  =  5645.376  Square  Varas. 

Reduce  to  square  varas : 

1.  1.77136  A.  3.  5  labors.  5.  2  sq.  leagues. 

2.  5314.08  A.  4.   10  labors.  6.   19  sq.  leagues. 

Reduce  to  acres : 

7.  5000000  sq.  varas.  9.   5  sq.  leagues.         11.   250  labors. 

8.  16936.18  sq.  varas.          10.   15  labors.  12.   25  sq.  leagues. 


MEASURES   OF   CAPACITY.  421 

MEASURES   OP   CAPACITY. 
LIQUID   MEASURE. 

594.  Liquid  Measure  is  used  in  measuring  liquids. 

TABLE. 

4  Gills  (gi.)  .=  1  Pint  .  .  .  pt. 

2  Pints          =  1  Quart  .  .  .  qt. 

4  Quarts       =  1  Gallon  .  .  .  gal. 

gal.    qt.     pt.      gi. 

1  =  4  =  8  =  32 

Scale.—  4,  2,  4. 

1.  In  determining  the  capacity  of  cisterns,  reservoirs,  etc.,  31£  gal- 
lons are  considered  a  barrel  (bbl.),  and  2  barrels,  or  63  gallons,  a 
hogshead  (hhd.).     In  commerce,  however,  the  barrel  and  hogshead  are 
not  fixed  measures. 

2.  Casks  of  large  size,  called  tierces,  pipes,  butts,  tuns,  etc.,  do  not 
hold  any  fixed  quantity.     Their  capacity  is  usually  marked  upon  them. 

3.  The  standard  gallon  of  the  United  States  contains  231  cubic 
inches,  and  will  hold  a  little  over  8J-  Ib.  of  distilled  water.     The  im- 
perial gallon,  now  adopted  by  Great  Britain,  contains  277.274  cu.  in., 
or  10  Ib.  of  distilled  water,  temperature  62°  Fahr.,  the  barometer  stand- 
ing at  30  inches. 

4.  The  beer  gallon  is  not  now  in  use.    It  contained  282  cubic  inches. 

DRY   MEASURE. 

595.  Dry   Measure   is   used   in   measuring   grain,    roots, 

fruit,  etc. 

TABLE. 

2  Pints  (pt.)  =  1  Quart  .  .  .  qt. 
8  Quarts  =  1  Peck  .  .  .  pk. 
4  Pecks  =  1  Bushel  .  .  .  bu. 

bu.    pk.      qt.       pt. 

1  =  4  =  32  =  64 

Scale.  —  4,  8,  2. 

1.  In  measuring  grain,  seeds,  or  small  fruits,  the  measure  must  be 
even  full  or  stricken.    In  measuring  large  fruits  or  coarse  vegetables, 
corn  in  the  ear,  etc.,  the  measure  should  be  heaped  at  least  six  inches. 

2.  Five  stricken  bushels  are  considered  equal  to  4  heaped  bushels. 
The  stricken  bushel  is  now  little  used,  except  to  ascertain  capacities. 


422          TABLES  OF   DENOMINATE   NUMBERS. 

3.  The  Winchester  bushel  is  the  standard  unit  in  the  United  States. 
In  form  it  is  a  cylinder,  18^  in.  in  diameter,  and  8  in.  deep,  and  con- 
tains 2150.42  cu.  in.     The  Winchester  bushel  was  discarded  by  Great 
Britain  in  1826,  and  the  imperial  bushel  was  substituted.    It  contains 
2218.192  cu.  in. 

4.  A  pint,  quart,  or  gallon,  dry  measure  is  more  than  the  same 
quantity,  liquid  measure ;  for  a  quart,  dry  measure,  is  -^  of  a  bushel, 
or  -fa  of  2150.4  cu.  in.,  which  is  about  67£  cu.  in.,  while  a  quart  liquid 
measure  is  \  of  231  cu.  in.,  or  57  f  cu.  in. 

Cu.  In.  Cu.  In.  Cu.  In.  Cu.  In. 

in  One  Gal.         in  One  Qt.        in  One  Pt.        in  One  Gi. 

Liquid  Meas.   231       67|      28J      7^ 
Dry  Meas.    268f      67£      33f      8f 

MEASURES   OP   WEIGHT. 
AVOIRDUPOIS    WEIGHT. 

596.  Avoirdupois  Weight  is  used  in  measuring  all  coarse 
and  heavy  articles,  as  hay,  grain,  groceries,  coal,  etc.,  and 
the  metals,  except  gold  and  silver. 

TABLE. 

16  Ounces  (oz.)        =  1  Pound  .    .     .     .     .  Ib. 

100  Pounds  =  1  Hundred-weight     .  cwt. 

20  Hundred-weight  =  1  Ton T. 

T.        CWt.  Ib.  OZ. 

1  =  20  =  2000  =  32000 
Scale.— 20,  100,  16. 

1.  In  weighing  coal  at  the  mines  and  in  levying  duties  at  the  United 
States  Custom  House,  the  long  ton  of  2240  Ib.  is  sometimes  used. 

2.  The  ounce  is  considered  as  16  drams. 

3.  The  unit  is  the  pound.    It  contains  7000  grains. 

The  following  denominations  are  also  used  : 
14  Ib.  =  1  Stone. 

100  Ib.  Butter  =  1  Firkin. 

100  Ib.  Grain  or  Flour  =  1  Cental. 

100  Ib.  Dried  Fish  =  1  Quintal. 

100  Ib.  Nails  =  1  Keg. 

196  Ib.  Flour  =  1  Barrel. 

200  Ib.  Pork  or  Beef  =  1  Barrel. 

280  Ib.  Salt  at  N.  Y.  Works  =  1  Barrel. 


MEASURES  OF   WEIGHT.  423 

In  states  that  regulate  the  weight  of  a  bushel,  the  follow- 
ing are  the  statutory  weights : 

Wheat,  60  Ib.     Eye,  66  lb.,  except  in  Cal.  54  Ib.  ;  in  La.  32  Ib. 

Corn,  shelled,  56  lb.,  except  in  Cal.  52  lb.  Corn  in  the  ear,  70  lb., 
except  in  Miss.  72  lb. ;  in  O.  68  lb.  ;  in  Ind.  after  Dec.  I  and  in  Ky. 
after  May  1,  following  the  time  of  husking  it,  68  lb. 

Oats,  32  lb.,  except  in  Ida.  and  Or.  36  lb.  ;  in  Md.  26  lb. ;  in  N.  J. 
and  Va.  30  lb.  Barley,  48  lb.,  except  in  Or.  46  lb. ;  in  Ala.,  Ga.,  Ky., 
Pa.,  47  lb. ;  in  Cal.  50  lb. ;  in  La.  32  lb. 

Buckwheat,  52  lb.,  except  in  Cal.  40  lb. ;  in  Conn.,  Me.,  Mass., 
Mich.,  Miss.,  N.  Y.,  Pa.,  Vt.,  Wis.,  48  lb. ;  in  Ida.,  N.  D.,  Okl.,  Or., 
S.  D.,  Tex.,  Wash.,  42  lb. ;  in  Kan.,  Minn.,  N.  J.,  N.  C.,  O.,  Tenn., 
50  lb. ;  in  Ky.  56  lb. 

Clover  seed,  60  lb.,  except  in  N.  J.  64  lb.  Timothy  seed,  45  lb., 
except  in  Ark.  60  lb. ;  in  N.  D.,  S.  D.,  42  lb. 

Bran,  20  lb.  Corn  meal,  50  lb.,  except  in  Ala.,  Ark.,  Ga.,  111., 
Miss.,  N.  C.,Tenn.,  48  lb.  Potatoes,  60  lb.,  except  in  Md.,  Pa.,  Va., 
56  lb.  Coal,  80  lb.,  except  in  Ky.,  Pa.,  76  lb. 

Peas,  60  lb.    Beans,  60  lb.,  except  in  Me.  62  lb. ;  in  Mass.  70  lb. 

1  TROY  WEIGHT. 
597.   Troy  Weight  is  used  in  weighing  gold,  silver,  and 

jewels. 

TABLE. 

24  Grains  (gr.)     =  1  Pennyweight    .     .    .     pwt. 

20  Pennyweights  =  1  Ounce       oz. 

12  Ounces  =  1  Pound lb. 

lb.       oz.      pwt.         gr. 
1  =  12  =  240  =  5760 

Scale.  — 12,  20,  24. 

1.  In  weighing  diamonds,  pearls,  and  other  jewels,  the  unit  com- 
monly employed  is  the  carat,  which  is  equal  to  4  carat  grains,  or  3.168 
Troy  grains. 

2.  The  term  carat  is  also  used  to  express  the  fineness  of  gold,  and 
means  ^  part.     Thus,  gold  that  is  18  carats  fine  is  ££  gold,  and  fa 
alloy. 

3.  The  standard  unit  of  weight  is  the  Troy  pound.     It  is  equal  to 
the  weight   of  22.7944  cu.  in.   of   distilled  water  at  its  maximum 
density,  the  barometer  being  at  30  inches.     It  is  identical  with  the 
Troy  pound  of  Great  Britain. 


424          TABLES  OF  DENOMINATE   NUMBERS. 

APOTHECARIES'   WEIGHT. 

598.  Apothecaries'  Weight  is  used  by  apothecaries  and 
physicians  in  weighing  medicines  for  prescriptions. 

TABLE. 

20  Grains  (gr.)  =  1  Scruple     .  .     .     sc.,  or  3 

3  Scruples         =  1  Dram  .     .  .     .     dr.,  or  3 

8  Drams  =  1  Ounce      .  .     .     oz. ,  or  5 

12  Ounces  =  1  Pound      .  .     .     lb.,  orft 

lb.     oz.      dr.         BC.          gr. 

1  =  12  =  96  =  288  =  5760 

Scale.  — 12,  8,  3,  20. 

1.  In  writing    prescriptions,   physicians  express    the  number  in 
Roman  characters,  using  j  instead  of  i  final.     They  also  write  the 
symbol  first ;  thus:    §v>  3vj,  3ij. 

2.  Medicines  are  bought  and  sold  in  large  quantities  by  Avoirdu- 
pois weight. 

1  lb.  Avoirdupois  =  7000  gr.     1  lb.  /  Trov  ^nd         \  =  5760  gr. 

I  Apothecaries'  / 

1  oz.  "  =  437|  gr.     1  oz.  "  =    480  gr. 

3.  It  will  be  observed  that  the  pound  is  identical  with  the  Troy 
pound,  as  are  also  the  ounce  and  the  grain,  though  the  ounce  is 
differently  divided. 

APOTHECARIES'  LIQUID  MEASURE. 

599.  Apothecaries'  Liquid  Measure  is  used  in  compounding 
and  measuring  liquid  medicines. 

TABLE. 

60  Drops  (gtt.)  or  minims  (n\,)  -  1  Fluid  drachm     .     .  /3. 

8  Fluid  drachms  =  1  Fluid  ounce    .     .     .  /J. 

16  Fluid  ounces  =  1  Pint       0. 

8  Pints  =  1  Gallon Cong. 

Cong.  O.    /?.      /3-          m. 
1  =  8  =  128  =  1024  =  61440 
Scale.  —  8,  16,  8,  60. 

The  abbreviation  Cong,  is  from  the  Latin  congius,  a  gallon.  A 
pint  being  one  eighth  of  a  gallon  the  abbreviation  for  it  is  O.,  from  the 
Latin  octarius,  one  eighth. 


MEASURES  OF   TIME.  425 

MEASURES  OP  TIME. 
600.   The  following  are  the  ordinary  divisions  of  time : 

min. 
hr. 
da. 
wk. 


60  Seconds  (sec.)  = 

Minute     .     . 

60  Minutes             = 

Hour  .     .     . 

24  Hours 

Day     .     .     . 

7  Days                  = 

Week      .     . 

365  Days 

Year   .     .     . 

366  Days                  =  ] 

L  Leap  Year    . 

100  Years                 =  1 

I  Century  .    . 

r.        mo.           da. 

hr.                min. 

cen. 

sec. 
1  =  100  =  1200  =  36500  =  876000  =  52560000  =  3153600000. 

Scale.  — 100,  365,  24,  60,  60. 

1.  In  most  business  computations  30  days  are  considered  a  month, 
and  12  months  a  year.    For  many  purposes  4  weeks  constitute  a  month. 

2.  The  common  year  contains  52  weeks  and  1  day,  the  leap  year  52 
weeks  and  2  days.     Hence,  commonly,  each  year  begins  one  day  later 
in  the  week  than  did  the  preceding  year,  but  the  year  succeeding  leap 
year  begins  two  days  later. 

3.  The  time  required  for  the  earth  to  revolve  around  the  sun  is  one 
year,  which  is  365  days  5  hr.  48  min.  49.7  sec.,  or  very  nearly  365|- 
days.    Instead  of  reckoning  this  part  of  a  day  each  year,  it  is  disre- 
garded, and  an  addition  is  made  when  this  amounts  to  one  day,  which 
is  very  nearly  every  fourth  year.     This  addition  of  one  day  is  made 
to  the  month  of  February.     Since  the  part  of  a  day  that  is  disre- 
garded when  365  days  are  considered  as  a  year,  is  a  little  less  than 
one  quarter  of  a  day,  the  addition  of  one  day  every  fourth  year  is  a 
little  too  much,  and,  to  correct  this  excess,  addition  is  made  to  only 
every  fourth  centennial  year.     With  this  correction  the  error  does  not 
amount  to  much  more  than  a  day  in  4000  years.     Therefore, 

Centennial  years  exactly  divisible  by  400,  and  other  years 
exactly  divisible  by  4,  are  Leap  Years. 

4.  The  reckoning  of  time  among  the  ancients  was  very  inaccurate. 
This  was  owing  to  their  ignorance  of  astronomy,  and  also  to  changes 
that  were  made  from  time  to  time  for  political  reasons.     The  calendar 
was  reformed  by  Julius  Csesar,  46  B.C.,  who  made  the  year  consist 
of  365|  days,  adding  one  day  every  fourth  year.    In  1582,  the  error 


426          TABLES  OF  DENOMINATE   NUMBERS. 

in  the  calendar  established  by  him  had  increased  to  10  days  ;  that  is, 
too  much  time  had  been  reckoned  as  a  year,  until  the  civil  year  was  10 
days  behind  the  solar  year.  To  correct  this  error.  Pope  Gregory  XIII. 
decreed  that  10  days  should  be  stricken  from  the  calendar,  that  the 
day  following  the  3d  day  of  October,  1582,  should  be  made  the  14th, 
and  that  henceforth  only  those  centennial  years  should  be  leap  years 
which  are  divisible  by  400. 

5.  Most  Catholic  countries  adopted  the  Gregorian  Calendar  soon 
after  it  was  established.     Great  Britain  did  not  adopt  it  until  1752, 
when  the  error  amounted  to  11  days.    By  Act  of  Parliament,  the  3d 
of  September  was  called  the  14th.    The  civil  year  by  the  same  act  was 
made  to  commence  on  the  1st  of  January,  instead  of  the  25th  of  March, 
as  was  previously  the  case. 

6.  Dates  reckoned  by  the  Julian  calendar  are  called  Old   Style 
(O.S.),  and  those  reckoned  by  the  Gregorian  calendar  are  called  New 
Style  (N.S.).     Russia  still  reckons  dates  according  to  Old  Style.     The 
difference  now  amounts  to  12  days. 

The  year  begins  with  the  month  of  January,  and  ends 
with  the  month  of  December. 

The  months,  their  names,  and  the  number  of  days  in  each, 
are  as  follows : 

.  July. 
.  Aug. 
.  Sept. 
.  Oct. 
.  Nov. 
Dec. 


CIRCULAR   OR   ANGULAR   MEASURE. 

601.  Circular  or  Angular  Measure  is  used  to  measure  arcs 
of  circles  and  angles,  in  determining  latitude,  longitude, 
direction,  the  position  of  vessels  at  sea,  etc. 

602.  That  part  of  the  circumference  which  is  included 
between  the  lines  which  form  the  angle  is  the  Measure  of 
the  Angle. 


January, 

31  da.     .     . 

Jan. 

July,            31  da. 

February, 

28  or  29  da. 

Feb. 

August,       31  da. 

March, 

31  da.     .     . 

Mar. 

September,  30  da. 

April, 

30  da.     .     , 

Apr. 

October,       31  da. 

May, 

31  da.     .     . 

May. 

November,  30  da. 

June, 

30  da.     .     . 

June. 

December,'  31  da. 

MEASURES  OF  VALUE.  427 

TABLE. 

60  Seconds  (")  =  1  Minute    .     .     .     ' 
60  Minutes        =  1  Degree     .     .     .     ° 
360  Degrees         =  1  Circumference  .     Cir. 
Cir.       o  •  " 

1  =  360  =  21600  =  1296000 
Scale.  —  360,  60,  60. 

1.  A  Quadrant  is  $  of  a  circumference,  or  90°  ;  a  Sextant  is  £  of  a 
circumference,  or  60°. 

2.  The  length  of  a  degree  of  longitude  on  the  earth's  surface  at  the 
Equator  is  69.16  miles. 

3.  In  astronomical  calculations  30°  are  called  a  Sign,  and  there 
are  therefore  12  signs  in  a  circle. 

MEASURES  OP  VALUE. 

603.  The  common  measure  of  value  is  Money. 

It  is  also  called  Currency,  and  is  of  two  kinds,  viz. :  coin  and  paper 
money. 

604.  Stamped  pieces  of  metal  having  a  value  fixed  by 
law  are  Coin  or  Specie. 

605.  Notes   and   bills   issued  by  the   Government   and 
banks,   and   authorized  to   be   used   as  money,  are  Paper 
Money. 

606.  All  moneys  which,  if  offered,  legally  satisfy  a  debt 
are  a  Legal  Tender. 

UNITED  STATES  MONEY. 

607.  The  unit  of  United  States  or  Federal  money  is  the 
Dollar. 

TABLE. 

10  Mills  (m.)  =  1  Cent    .     .     ct.     I      10  Dimes    =  1  Dollar.     .     $ 
10  Cents         =  1  Dime  .     .     d.       I      10  Dollars  =  1  Eagle  .     .     E. 

Scale.  — Decimal. 

1.   The  dollar  mark  is  probably  a  combination  of  U.  S.,  the  initials 
of  the  words  "  United  States." 


428  TABLES  OF  DENOMINATE  NUMBERS. 

2.  The  coins  of  the  United  States  are  — 

Gold :  The  double-eagle,  eagle,  half-eagle,  quarter-eagle,  and  one- 
dollar  piece. 

Silver:  The  dollar,  half-dollar,  quarter-dollar,  and^the  ten-cent 
piece. 

Nickel :  The  five-cent  piece. 
Bronze  :  The  one-cent  piece. 

There  are  various  other  coins  of  the  United  States  hi  circulation, 
but  they  are  not  coined  now. 

The  denominations  dimes  and  eagles  are  rarely  used,  the  dimes 
being  regarded  as  cents,  and  the  eagles  as  dollars. 

3.  The  unit  of  value  is  the  dollar.     Its  standard  weight  in  gold  is 
25.8  gr. 

4.  The  standard  purity  of  the  gold  and  silver  coins  is  by  weight  9 
parts  of  pure  metal  and  1  part  alloy. 

The  alloy  of  gold  coins  consists  of  silver  and  copper ;  the  silver,  by 
law,  is  not  to  exceed  one  tenth  of  the  alloy. 

The  alloy  of  silver  coins  is  pure  copper. 

The  nickel  coins  consist  of  one  fourth  nickel  and  three  fourths 
copper. 

The  cent  is  composed  of  95  parts  copper  and  5  parts  tin  and  zinc. 

5.  All  gold  coins  are  a  legal  tender  for  any  amount ;  silver  coins 
less  than  $  1  are  legal  tender  for  any  amount  not  exceeding  $  10  in  any 
one  payment ;  nickel  and  bronze  corns,  for  any  amount  not  exceeding 
25  cents  in  any  one  payment. 


CANADA    MONEY. 

• 

608.  The  currency  of  Canada  is  decimal,  and  the  table  and 
denominations  are  the  same  as  those  of  United  States  money. 
English  money  is,  however,  still  used  to  some  extent. 

1.  The  coins  of  Canada  are  for  the  most  part  of  the  same  denomi- 
nations as  those  of  the  United  States,  except  the  gold  coins,  which  are 
the  sovereign  and  half-sovereign. 

2.  Canadian  coins  are  not  received  at  their  full  face  value  in  many 
parts  of  the  United  States.    The  half-dollar  pieces  are  taken  for  only 
40  cents,  the  quarter-dollar  pieces  for  only  20  cents,  etc. 


MEASURES   OF   VALUE.  429 

ENGLISH  OR  STERLING  MONEY. 

609.  English  money  is  the  currency  of  Great  Britain. 
The  unit  is  the  Pound  or  Sovereign. 

TABLE. 

4  Farthings  (far.)  =     1  Penny .     .     .     d. 
12  Pence  =     1  Shilling     .     .    s. 

20  Shillings  =  /I  ^nd,  or  I  .     £ 

1 1  Sovereign  J 

£       s.         d.         far. 

1  =  20  =  240  =  960 

Scale.  —  20,  12,  4. 

1.  Farthings  are  commonly  written  as  fractions  of  a  penny.    Thus, 
7  pence  3  farthings  is  written  7|d.  ;  5  pence  1  farthing,  5Jd. 

2.  The  value  of  £1  or  sbvereign  is  $4.8665  in  American  gold,  and 
the  other  coins  have  their  proportionate  values. 

3.  The  coins  of  Great  Britain  in  general  use  are  — 

Gold  :  Sovereign,  half-sovereign,  and  guinea,  which  is  equal  to  21 
shillings. 

Silver:  The  crown  (equal  to  5  shillings),  half-crown,  florin  (equal 
to  2  shillings),  shilling,  six-penny  and  three-penny  pieces. 

Copper :  Penny  and  half -penny. 

FRENCH  MONEY. 

610.  In  France  the  currency  is  decimal.     The  unit  is  the 
Franc. 

TABLE. 

10  Centimes  (ct. )  [pronounced  son-teems']  =  1  Decime    .     .     do. 

10  Decimes  [pronounced  des-seems]  =  1  Franc  .     .     .     fr. 

Scale.  —  Decimal. 

1.  The  value  of  the  franc,  as  determined  by  the  Secretary  of  the 
Treasury,  is  $ .  193  in  United  States  money. 

2.  The  coins  of  France  are  of  gold,  silver,  bronze,  and  copper. 
The  gold  coins  are  the  hundred,  forty,  twenty,  ten,  and  Jive  franc 
pieces ;  the  silver  coins  are  the  Jive,  two,  and  one  franc  pieces ;  also 
the  fifty  and  twenty-five  centime  pieces.     The  bronze  coins  are  the  ten, 
Jive,  two,  and  one  centime  pieces*     There  are  also  copper  coins  in  ten 
and  ./fog  centime  pieces. 


430          TABLES   OF  DENOMINATE  NUMBERS. 


GERMAN  MONEY. 

611.  German  money  is  the  legal  currency  of  the  German. 
Empire. 

TABLE. 

100  Pfennigs  =  1  Mark. 
Scale.  —  Decimal. 

1.  The  unit  is  the  mark.     Its  value  is  $.2385  in  United  States 
money. 

2.  The  coins  of  the  German  Empire  are  of  gold,  silver,  nickel,  and 
copper.     The  gold  coins  are  the  20-mark  piece,  the  10-mark  piece, 
and  the  5-mark  piece.     The  silver  coins  are  the  two  and  one  mark 
pieces  ;  the  nickel  coins  are  the  ten  and  five  pfennig  pieces  ;  and  the 
copper  coins  are  the  two  and  one  pfennig  pieces. 

COUNTING. 

612.  The  following  denominations  are  used  in  counting 
some  classes  of  articles  : 

TABLE. 

12  Things  =  1  Dozen doz. 

12  Dozen  =  1  Gross gr. 

12  Gross     =  1  Great  Gross     .     .     .     G.  gr. 

Scale.  —  Duodecimal. 

Two  things  are  often  called  a  pair,  and  twenty  things  a  score  ;  as  a 
pair  of  birds,  a  score  of  years. 

STATIONERS'  TABLE. 

613.  The  denominations  used  in  the  paper  trade  are  : 


24  Sheets     =  1  Quire. 
20  Quires    =  1  Ream. 


2  Reams    =  1  Bundle. 
5  Bundles  =  1  Bale. 


Bale    Bundles    Reams      Quires       Sheets 
1    =    5=    10    =    200  =  4800 

Scale.—  5,  2,20,24. 

The  terms  folio,  quarto,  octavo,  applied  to  books,  indicate  the 
number  of  leaves  into  which  a  sheet  of  paper  is  folded.  Thus,  when 
a  sheet  of  paper  is  folded  into  2,  4,  8,  12,  16,  18,  or  24  leaves,  the 
forms  are  called  respectively,  folio,  4to  or  quarto,  8vo  or  octavo,  12mo, 
16mo,  18mo,  and  24mo. 


VERMONT  RULES.  431 

INTEREST  AND  PARTIAL  PAYMENTS. 
VERMONT  RULES. 

614.  The  Vermont  statutes  contain  the  following  provisions 
regarding  the  computation  of  time  and  rates  of  interest : 

SEC.  12.  The  word  " month  "  shall  mean  a  calendar  month ;  and  the  word  "year  " 
shall  mean  a  calendar  year, ... 

SEC.  26.  When  time  is  to  be  reckoned  from  a  day  or  date,  or  an  act  done,  such  day, 
date,  or  day  when  such  is  done  shall  not  be  included  in  the  computation. 

SEC.  2301.  The  rate  of  interest,  or  the  sum  allowed  for  the  forbearance  or  use  of 
money,  shall  be  six  dollars  for  one  hundred  dollars  for  one  year,  and  at  the  same  rate 
for  a  greater  or  less  sum,  and  for  a  longer  or  shorter  time, ... 

The  rule  and  custom  in  Vermont,  in  commercial  transactions,  in 
the  banks,  and  in  the  courts,  in  computing  interest,  is  to  consider  a 
calendar  month  whether  it  contains  28,  29,  30,  or  31  days  as  one 
twelfth  of  a  year ;  a  day,  or  any  number  of  days  up  to  thirty,  as  so 
many  thirtieths  of  a  month. 

When  it  is  required  to  find  the  time  between  two  dates  less  than  a 
calendar  month  apart,  count  the  actual  number  of  days. 

Thus,  from  January  28,  1895,  to  February  1,  1895,  is  four  days ;  from  February  28, 
1895,  to  March  1, 1895,  is  one  day ;  from  February  28,  1892,  to  March  1,  1892,  is  two 
days,  February  in  that  year  having  29  days. 

The  custom  and  the  law  relating  to  the  time  when  a  note  matures 
and  to  the  amount  due  at  its  maturity  are  illustrated  by  the  following 


A  note  dated  January  28, 1895,  for  $100,  payable  one  month  after  date,  with  interest, 
became  payable  February  28, 1895  (31  days  after  date),  and  the  amount  due  was  $  100.50. 

A  note  dated  January  29, 1895,  for  $  100,  payable  one  month  after  date,  with  interest, 
became  payable  February  28, 1895  (30  days  after  date),  and  the  amount  due  was  $  100.50. 

A  note  dated  January  30,  1895,  for  $100,  payable  one  month  after  date,  with  inter- 
est, became  payable  February  28,  1895  (29  days  after  date),  and  the  amount  due  was 
$  100.50. 

A  note  dated  January  31, 1895,  for  $  100,  payable  one  month  after  date,  with  interest, 
became  payable  February  28, 1895  (28  days  after  date),  and  the  amount  due  was  $  100.50. 

A  note  dated  January  31,  1895,  for  $100,  payable  30  days  after  date,  with  interest, 
became  payable  on  March  2,  1895,  and  the  amount  due  was  $  100.50. 

A  note  was  dated  January  1,  1895,  for  $100,  payable  on  demand,  with  interest. 
It  was  paid  January  31,  1895 ;  the  amount  due  was  $  100.50,  (30  days). 

A  note  was  dated  January  1,  1895,  for  $  100,  payable  on  demand,  with  interest. 
It  was  paid  February  1,  1895 ;  the  amount  due  was  $  100.50,  (a  calendar  month). 


432         INTEREST  AND   PARTIAL  PAYMENTS. 

A  note  was  dated  February  1,  1895,  for  $  100,  payable  on  demand,  with  interest. 
It  was  paid  February  28,  1895 ;  the  amount  due  was  $  100.45,  (27  days). 

A  note  was  dated  February  1,  1895,  for  $  100,  payable  on  demand,  with  interest. 
It  was  paid  March  1,  1895 ;  the  amount  due  was  $  100.50,  (a  calendar  month). 

A  note  dated  January  1,  1895,  payable  31  days  after  date,  with  interest,  became 
payable  on  February  1,  1895,  and  the  holder  was  entitled  to  interest  for  one  month  and 
one  day. 

The  Vermont  statutes  contain  the  following  provisions  relating 
to  the  modes  of  computing  the  indebtedness  upon  notes,  bills,  or  other 
similar  obligations : 

FIRST,  When  the  note  or  debt  draws  simple  interest. 

SEC.  2302.  On  notes,  bills,  or  other  similar  obligations,  payable  on 
demand  or  at  a  specified  time,  with  INTEREST,  when  payments  are 
made,  such  payments  shall  be  applied :  first,  to  liquidate  the  interest 
accrued  at  the  time  of  such  payments;  and  second,  to  extinguish  the 
principal. 

[It  will  be  observed  that  this  rule  is  similar  to  the  United  States  Rule  (Art.  361).] 

SECOND,  When  the  note  or  debt  draws  annual  interest. 

SEC.  2303.  When  such  obligations  are  payable  on  demand  or  at  a 
specified  time,  with  INTEREST  ANNUALLY,  the  annual  interest  that  re- 
mains unpaid  shall  bear  simple  interest  from  the  time  it  becomes  due 
to  the  time  of  final  settlement ;  but  if  in  any  year,  reckoning  from  the 
time  such  annual  interest  began  to  accrue,  payments  are  made,  the 
amount  of  such  payments  at  the  end  of  such  year,  with  interest  thereon 
from  the  time  of  payment,  shall  be  applied :  first,  to  liquidate  the  simple 
interest  accrued  from  the  unpaid  annual  interest ;  second,  to  liquidate 
the  annual  interests  due  ;  and  third,  to  extinguish  the  principal. 

WRITTEN   EXERCISES. 

1.  $2000.  BARRE,  VT.,  July  15,  1885. 

On  demand,  for  value  received,  I  promise  to  pay  to  the  order  of 
William  D.  Hudson,  two  thousand  dollars,  with  interest  annually. 

SAMUEL  S.  SPURR. 

Indorsed  as  follows:  Dec.  10,  1886,  $500;  Aug.  15,  1887,  $20; 
Feb.  16,  1888,  $25;  Nov.  12,  1890,  $20  ;  Dec.  12,  1891,  $576.  How 
much  was  due  April  18,  1892  ? 


VERMONT  RULES. 


433 


SOLUTION. 


Principal  July  15, 1885, 

Int.  on  prin.  to  July  15,  1887  (2  yr.), 

Int.  on  $  120  yearly  int.  unpaid  for  1  yr., 

Total  int.  due  July  15,  1887, 

Ain't  of  1st  pay't  July  15, 1887 

(7  mo.  5  da.),  $517.92 

Bal.  applied  to  liquidate  principal 

($517.92-$247.20), 


(1) 

Int.  on  unpaid 
yearly  int.      year 


(2) 


Unpaid 
Ty  int 


(3) 

Principal*. 

$2000.00 


$240.00 


$7.20 


$247.20 


270.72 


Principal  (balance  due)  July  15, 1887, 
Int.  on  prin.  to  July  15,  1888  (1  yr.), 
Am'tof  2d  pay't  to  July  15, 1888(11  mo.), 
Am't  of  3d  pay't  to  July  15, 1888  (5  mo.), 
Sum  of  pay'ts  applied  to  liquidate  yearly 

int., 
Bal.  of  unpaid  yearly  int.  July  15,  1888, 


$  1729.28 


$21.10 
25.63 


$  103.76 


46.73 
57.03 


Principal  July  15,  1888, 

Int.  on  prin.  to  July  15,  1891  (3  yr.), 

Int.  on  $  103.76,  the  yearly  int.  on  prin. 

for  (2  yr.  + 1  yr.)  3  yr., 
Yearly  int.  hitherto  unpaid, 
Int.  on  bal.  of  unpaid  yearly  int.,  $57.03, 

for  3  yr., 

Total  int.  upon  unpaid  yearly  int. 
Total  unpaid  yearly  int., 
Am't  of  4th  payment  to  July  15,  1891 

(8  mo.  3  da.),  applied  to  liquidate 

int.  upon  unpaid  yearly  int., 
*  Bal.  of  int.  upon  unpaid  yearly  int., 


$  1729.28 


$18.68 


10.27 


$311.27 


57.03 


368.30 


20.81 

8.14 


Principal  July  15,  1891, 

Int.  on  prin.  to  April  18, 1892  (9  mo.  3  da.), 

Unpaid  yearly  int.  July  15, 1891, 

Int.  upon  $  368.30  unpaid  yearly  int.  to 

April  18, 1892, 

*  Bal.  of  int.  upon  unpaid  yearly  int., 
Total  int.  due  upon  unpaid  yearly  int., 
Total  int.  due  at  time  of  settlement, 
Am't  of  note  at  time  of  settlement, 
Am't  of  pay't  to  Apr.  18, 1892  (4  mo.  6  da.), 
Sum  due  Apr.  18,  1892, 


$  1729.28 


$16.76 
8.14 


$78.68 
368.80 


24.90 


471.88 

2201.16 

587.08 

$  1614.08 


*  No  interest  is  paid  upon  the  interest  upon  unpaid  yearly  interest. 

2.  A  note  dated  June  6,  1890,  was  given  for  $  2250,  with  annual 
interest  at  6  %.  Aug.  10,  1892,  a  payment  of  $  1000  was  indorsed. 
What  amount  was  due  Jan.  1,  1894  ? 


434  NEW  HAMPSHIRE   RULE. 

3.  $800.  BENNINGTON,  Vx.,  Jan.  30,  1888. 
On  demand,  I  promise  to  pay  the  bearer  eight  hundred  dollars,  for 

value  received,  with  interest  annually  at  6%.  CHAS.  H.  SMITH. 

The  interest  on  the  above  note  was  regularly  paid  and  indorsed  for 
3  years.  What  was  due  Jan.  30,  1894  ? 

4.  A  note  for  $  1000,  given  Oct.  5,  1891,  bearing  annual  interest  at 
6%,  was  indorsed  as  follows:  Dec.  17,  1891,  $100;  June  26,  1892, 
$200  ;  Nov.  14,  1893,  $150.     What  remained  due  Jan.  1,  1895  ? 

5.  At  Rutland,  Vt.,  a  note  for  $500  was  given  Sept.  10,  1886,  to 
be  paid  on  demand,  interest  annually  at  6  %.    The  following  indorse- 
ments were  made:  April  1,  1888,  $20  ;  July  1,  1889,  $25 ;  March  19, 
1891,  $200.     What  amount  remained  due  after  the  last  payment  ? 

6.  Find  the  amount  due  July  1,  1889,  on  a  note  for  $  1250,  dated 
July  1,  1885,  annual  interest  6%,  and  indorsed  as  follows:  March  6, 
1887,  $250;  June  1,  1888,  $400;  Dec.  13,  1888,  $50;  May  1,  1889, 
$125. 

7.  A  note  of  $  1500,  payable  on  demand  with  6%  interest,  payable 
annually,  given  Feb.  1,  1880,  was  indorsed  as  follows:  Jan.  1,  1884, 
$100;  Jan.  13,  1886,  $40;  Oct.  1,  1886,  $900;  Feb.  1,  1888,  $400. 
What  amount  remained  due  after  the  last  payment  ? 

8.  Find  the  amount  due  on  the  following  note  July  1,  1891 : 
$3000.  VERGENNES,  VT.,  April  1,  1888. 

On  demand,  I  promise  to  pay  to  the  bearer  three  thousand  dollars, 
for  value  received,  interest  annually  at  6%.  THOMAS  HALL. 

Indorsements:  Nov.  5,  1888, '$300;  Jan.  4,  1890,  $100;  Aug.  20, 
1890,  $  1000. 

NEW   HAMPSHIRE   RULE. 

The  law  of  New  Hampshire  for  computing  the  indebtedness  upon 
a  note  or  other  obligation  when  partial  payments  have  been  made  is 
the  same  as  the  Vermont  rule,  with  the  following  additional  provision : 

If,  however,  at  the  date  of  any  payment  there  is  no  interest  except 
the  accruing  annual  interest,  and  the  payment,  or  payments,  do  not 
exceed  the  annual  interest  at  the  end  of  the  year,  deduct  the  payment^ 
or  payments,  without  interest  on  the  same. 


INTEREST  AND  PARTIAL  PAYMENTS.  435 

1.  On  a  note  for  $2500,  given  at  Nashua,  N.H.,  Oct.  1,  1888, 
with  interest  payable  annually  at  6  %,  the  following  payments  were 
made:  June  1,  1890,  $500;  Mar.  17,  1891,  $87.94;  Dec.  1,  1893, 
$  1000.  How  much  was  due  Apr.  1,  1895  ? 

SOLUTION. 

Int.  on  unpaid    Unpaid      P-JT,,,?,,.!. 
yearly  in!      yearly  int.    ™xap«* 

Principal,  $2500.00 

Int.  on  prin.  to  Oct.  1,  1890  (2  yr.),  $  800.00 

Int.  on  $150,  yearly  int.  on  prin.  for  1  yr.,  $9.00 

Total  int.  due  Oct.  1,  1890,  $309.00 

Amt.  of  1st  paym't  to  Oct. 

1,1890,  $510.00 

Bal.  applied  to  liquidate  principal 

($510.00  —  $809.00),  201.00 


Principal  Oct.  1,  1890, 
Int.  on  prin.  to  Oct.  1,  1891, 
Second  pay't  without  int.,  because  less 
than  int.  due,  applied  to  liquidate 
int., 
Unpaid  yearly  int.  Oct.  1,  1891, 

$137.94 

87.94 
50.00 

$  2299.00 

Principal  Oct.  1,  1891,  $  2299.00 

Int.  on  prin.  to  Oct.  1,  1894  (8  yr.),  $418.82 

Int.  on  $  137.94,  the  yearly  int.  on  prin. 

for(2yr.+lyr.)8yr.,  $24.88 

Unpaid  yearly  int.  Oct.  1,  1891,  50.00 

Int.  on  unpaid  yearly  int.  to  Oct.  1,  1894 

(3  yr.),  9.00 

Total  unpaid  yearly  int.  _ 

Total  int.  upon  unpaid  yearly  int.,  $33.88 

Total  int.  due  Oct.  1,  1894,  $  497.65 

Am't  of  pay't  Oct.  1,  1894,  $  1050.00 

Bal.  applied  to  liquidate  the  principal 

($  1050.00  -  $  497.65),  $  552.35 

Principal  Oct.  1,  1894,  $  1746.65 

Int.  on  prin.  to  Apr.  1,  1895,  52.40 

Sum  due  Apr.  1,  1895,  $  1799.05 

2.  On  a  note  for  $2000,  given  at  Concord,  N.H.,  Apr.  1,  1883, 
with  interest  annually  at  6%,  the  following  payments  were  made: 
July  1,  1885,  $300;  June  1,  1886,  $35;  Dec.  1,  1887,  $500.     How 
much  was  due  Feb.  1,  1889  ? 

3.  On  a  note  for  $3600,  given  at  Manchester,  N.H.,  Feb.  10, 
1886,  with  interest  annually  at  6%,  the  following  payments  were 
made:   Dec.  10,  1886,  $100;  Aug.  10,  1888,  $600;  Sept.  10,  1890, 
$  350.     How  much  was  due  Mar.  10,  1892  ? 


436  CONNECTICUT   RULE. 


CONNECTICUT  RULE. 

The  Connecticut  court  rule  for  computing  the  indebtedness  upon 
a  note  or  other  obligation,  when  partial  payments  have  been  made,  is  : 

Compute  the  interest  to  the  time  of  the  first  payment,  if  that  be  one 
year  or  more  from  the  time  the  interest  commenced;  add  it  to  the  prin- 
cipal^ and  deduct  the  payment  from  the  sum  total. 

If  there  be  after  payments  made,  compute  the  interest  on  the  balance 
due  to  the  next  payment,  and  then  deduct  the  payment  as  above;  and 
in  like  manner  from  one  payment  to  another,  till  all  the  payments  are 
absorbed,  provided  the  time  between  one  payment  and  another  be  one 
year  or  more. 

But  if  any  payment  be  made  before  one  year's  interest  hath  accrued, 
then  compute  the  interest  on  the  principal  sum  due  on  the  obligation 
for  one  year,  add  it  to  the  principal,  and  compute  the  interest  on  the 
sum  paid,  from  the  time  it  was  paid,  up  to  the  end  of  the  year  /  add  it 
to  the  sum  paid,  and  deduct  that  sum  from  the  principal  and  interest 
added  as  above. 

If  any  payments  be  made  of  a  less  sum  than  the  interest  arisen  at 
the  time  of  such  payment,  no  interest  is  to  be  computed  but  only  on  the 
principal  sum  for  any  period. 

1.  On  a  note  for  $  650,  given  at  Hartford,  Conn.,  June  12, 1890,  with 
interest  at  6%,  the  following  payments  were  made:  July  1,  1891, 
$  116.20;  Apr.  10,  1892,  $61.50  ;  Feb.  12,  1893,  $12.10  ;  Aug. 20,  1893V 
$  110.  How  much  was  due  Dec.  21,  1893  ? 

SOLUTION. 

Principal $650.00 

Int.  to  July  1,1891  (lyr.  19  da.) 41.06 

Amount $691.06 

First  payment 116.20 

New  principal $574.86 

Int.  to  July  1, 1892  (second  payment  being  made  less  than  1  yr.  from  previous 

payment) 84.49 

Amount $609.35 

Am't  of  2d  pay'tfrom  Apr.  10  to  July  1,  1892  (2  mo.  21  da.)     ....  62.33 

New  principal $547.02 

Amount  to  July  1,  1893  (1  yr.) 579.84 

Third  payment  draws  no  interest,  being  less  than  int.  due         ....  12.10 

New  principal $567.74 

Amount  Dec.  21,  1898  (5  mo.  20  da.) 583.88 

Amount  of  last  payment  at  settlement  (4  mo.  1  da.) 112.22 

Balance  due  Dec.  21, 1893 4471.61 


TAXES.  437 

2.  On  a  note  for  $  1000,  given  at  New  Haven,  Conn.,  Feb.  1,  1890, 
with  interest  at  6%,  the  following  payments  were  made:    Apr.   1, 
1891,  $80  j  Aug.  1,  1891,  $30  j  Oct.  1,  1892,  $10  ;  Dec.  1, 1892,  $600  ; 
May  1,  1893,  $200.    How  much  was  due  Oct.  1,  1893  ? 

3.  On  a  note  for  $1000,  given  at  Middletown,  Conn.,  Mar.  9,  1890, 
with  interest  at  6%,  the  following  payments  were  made:   Nov.  19, 
1890,  $204;  Mar.  3,  1892,  $50;  June  15,  1893,  $600;  Nov.  1,  1893, 
$  85.     How  much  was  due  Jan.  1,  1894  ? 

TAXES. 

In  Vermont,  public  revenues  are  derived  from  taxes  laid  on  the  busi- 
ness of  certain  corporations,  such  as  railroads,  insurance  companies, 
savings  banks,  etc.,  from  certain  licenses  and  fees,  from  fines,  from 
collateral  inheritance  taxes,  and  largely  from  direct  taxes  laid  upon 
real  and  personal  property  and  polls. 

The  method  of  computing  direct  taxes  in  Vermont  varies  a  little 
from  that  adopted  in  many  states.  It  is  as  follows : 

1.  A  list  of  the  real  estate  and  personal  property,  together  with  a  poll  list,  which  is 
f  2  for  every  male  person  over  21  and  under  70  years  of  age,  is  made  by  the  persons 
authorized  by  law  to  appraise  the  property  of  the  town. 

2.  Soldiers  severely  wounded  in  the  Civil  War,  soldiers  of  the  Civil  War  honorably  dis- 
charged having  no  taxable  estate,  and  persons  actually  poor  are  exempt  from  a  poll  tax. 

3.  Heal  estate  used  for  public,  religious,  charitable,  and  educational  purposes  is  for 
the  most  part  exempt  from  taxation. 

4.  United  States  bonds  and  certain  other  stocks  and  bonds  are  exempt  from  taxa- 
tion ;  also  savings  bank  deposits  to  the  amount  of  $  1500. 

5.  Personal  property  is  exempt  from  taxation  to  an  amount  equal  to  the  excess,  if 
any,  of  the  owner's  debts  over  the  aggregate  amount  of  his  .holdings  that  are  exempt 
according  to  Note  4 ;  also  household  goods,  farming  tools,  libraries,  etc. 

The  sum  of  the  poll  list  and  1%  of  the  taxable  real  and  personal 
property  constitutes  what  is  called  the  Grand  List  of  the  town. 

The  sum  of  each  person's  poll  tax  and  1  %  of  the  value  of  his  taxable 
real  and  personal  property  constitutes  his  grand  list. 

The  state  taxes  are  fixed  by  the  Legislature ;  ordinary  county  taxes,  by  county 
judges.  A  town  tax  is  levied  by  vote  of  the  town,  a  village  tax  by  vote  of  the  village. 

Taxes  are  levied  at  a  certain  number  of  cents  on  each  dollar  of  the 
grand  list,  or  a  certain  per  cent  of  the  grand  list. 

Thus,  the  Legislature  of  1894  assessed  a  tax  of  12  cents  on  the  dollar  for  state 
purposes. 


438  TAXES. 

WRITTEN   EXERCISES. 

1.  In  the  city  of  Rutland  the  taxable  real  estate  was  $5,531,685, 
the  taxable  personal  property,   $2,135,288,  and  the  taxable  polls, 
2625.     What  was  the  grand  list  ? 

SOLUTION. 

1  %  of  $  5,531,685  =  $  55,316.85 
1  %  of  $  2,135,288=  21,352.88 
2625  polls  at  $  2  =  5.250.00 
Grand  List  =  $  81,919.73 

2.  The  tax  rate  for  the  city  of  Rutland  in  1894  was  as  follows  :  for 
state,  5  cents ;  for  state  schools,  5  cents  ;  for  state  roads,  5  cents  ; 
for  city  highways,  20  cents  ;  for  city  schools,  40  cents ;  for  general 
city  purposes,  65  cents ;  for  city  sinking-fund,  10  cents.     What  was 
the  tax  of  a  man  50  years  of  age  whose  personal  property  was  assessed 
at  $6250,  and  whose  real  estate  was  rated  at  $21,550? 

SOLUTION. 

1%  of  $6250  =  $  62.50 
1%  of  $21,550  =  215.50 
Poll  tax  =  2.00 

His  Grand  List  =  $  280.00 

The  sum  of  the  several  taxes  is  150  cents,  therefore  the  total  tax  rate  is  150  %  of  the 
grand  list,  and  the  man's  total  tax  was 

$150%  of  $280,  or  $420. 

3.  The  town  of  Benson  raised  a  town  tax  of  55  %,  a  state  school  tax 
of  12  %,  and  a  highway  tax  of  15  %  of  the  grand  list.     What  are  the 
taxes  of  a  man  55  years  of  age  whose  real  estate  is  appraised  at  $  4500 
and  his  personal  property  at  $  3000  ? 

4.  In  a  certain  school  district  a  tax  of  15  cents  on  a  dollar  of  the 
grand  list  is  to  be  raised  for  school  purposes.     What  is  the  amount  of 
B's  school  tax  who  owns  a  farm  valued  at  $  2500  ? 

5.  What  was  the  grand  list  of  the  town  of  Lincoln  for   1894,  if 
it  had  1025  taxable  polls,  real  estate  valued  at  $1,206,175,  and  per- 
sonal property  worth   $835,978?     If  the  town   raised   $20,295  by 
taxes,  what  was  the  tax  upon  one  dollar  of  the  grand  list  ? 

6.  What  is  the  tax  of  a  man  72  years  old,  living  in  Lincoln,  whose 
real  estate  is  valued  at  $  4225  and  personal  property  at  $  5000,  if  in 
addition  to  his  town  tax  he  pays  a  state  school  tax  of  10  %  of  his  grand 
list? 


ANSWERS. 


Page  30. —2.  2378.        ! 
7.    $29.95.         8.   3609. 
12.  $24.51.        13.  $30.64. 
17.  35,472.       18.  $266.53. 
22.    $454.49.        23.    $482. 

Page  31.— 24.  17,070. 
28.  $97.58.     29.  $44.085. 
33.  $117.234.       34.  28,338. 
38.  16,803.        39.  11,377. 
43.    14,420.        44.    8687. 

Page  32.  — 46.  15,817. 
50.  8750.         51.    8,021,463. 
54.  9,101,736,502.      55.  649,424. 

Page  33.— 58.  723,074,817. 


3.  1892.   4.  1711.   5.  3116.   6.  3328. 

9.  4007.    10.  $33.72. 

11.  $26.00. 

14.  27,755. 

15.  36,376. 

16.  36,712. 

19.  46,688. 

20.  60,582. 

21.  62,112. 

40. 

25.  26,273. 

26.  $  125.99. 

27.  $  176.64. 

30.  $252.558. 

31.  $50.814. 

32.  $  96.968. 

.   35.  16,289. 

36.  20,424. 

37.  48,699. 

40.  15,247. 

41.  11,773. 

42.  18,604. 

45.  17,275. 

47.  15,665. 

48.  13,152. 

49.  20,555. 

52.  103,283,772.  53.  1,028,363,547. 
56.  130,556,589.  57.  145,770,476. 
59.  135,103,556.  60.  107,823,882. 


61.  1,936,123.  62.  50,451,121.  63.  146,305,505.  64.  9089. 
65.  6894. 

Page  34.— 66.  75,688.  67.  97,024.  68.  $3200.  69.  $1381. 
70.  226,382  shingles  ;  106,400  laths.  71.  $  5,165,000.  72.  8774. 

Page  35.  — 73.  11,240.  74.  3269.  75.  $41,819.  76.  55,080. 
77.  251,025.  78.  89,081. 

Page  36.  —  79.  66,465.  80.  $2353.45,  Albany;  $4700.27,  New 
York;  $7053.72,  both.  81.11,160.  82.  14,425,  wheat ;  16,740,  corn ; 
31,165,  grain.  83.  106,618.  84.  $47,492. 

Page  37. —j85.  16,201.  86.197,680.  87.45,050.  88.212,482. 
89.  91,260  Ib.  cotton  ;  331  hhd.  sugar  ;  10,550  gal.  molasses. 

Page  43.  — 2.  322.    3.  144.    4.  133.    5.  164.    6.  122. 

7.  4444.   8.  2222.   9.  1441.   10.  1512.   11.  2721.   12.  $23.52. 
13.  $34.34.    14.  $32.35.    15.  $31.43.    16.  $32.21. 

Page  44. —17.  $31.31.       18.  $10.81.       19.  $14.43.       20.  $32.12. 

21.  $41.44.        22.  24,413.        23.  13,221.        24.  41,111.        25.  38,133. 
26.  31,155.        27.  27,312.        28.  42,253.        29.  14,542.        30.  46,112. 
31.    24,112.         32.   $220.          33.    $2344.          34.    1023.          35.    616. 
36.    $1113.         37.    2210.        38.    3121.        39.    $142.25. 

Page  45.  —  40.    $1123.25.        41.    $333.        42.    13,211. 
Page  47.  — 2.  487.      3.  494.      4.  283.      5.  167.      6.  304. 

8.  416.         9.   291.         10.   308. 

Page  48.  —  11.  276.        12.  167.        13.  248.        14.  482. 
16.    192.         17.    137.        18.   324.        19.  327.        20.  378. 

22.  426.        23.    196.        24.    92.        25.   343.        26.   385. 

439 


7.  119. 

15.  189. 
21.  189. 
27.  108. 


440 


ANSWERS. 


28.  290.       S 
34.  34,724. 
39.  10,960. 
44.   30,641. 
49.  28,507. 
64.  $451.60. 
59.176.98. 
64.    $60.46. 


220.       30.  168.       31.  385. 
35.  12,827.        36.  24,298. 
41.  17,987. 
46.  31,166. 
51.  $  195.45. 
56.  $  88.09. 
61.  $  504.28. 


40.  21,060. 

45.  19,531. 
50.  .$122.93. 

55.  $  489.78. 
60.  $455.79. 
65.    $264. 


32.  7889. 
37.  24,588. 
42.  40,516. 
47.  7200. 
52.  $196.79. 
57.  $371.76. 
62.  $247.74. 

33.  20,725. 
38.  52,583. 
43.  34,594. 
48.  14,207. 
53.  $  21.84. 
58.  $  154.75. 
63.  $  59.72. 

68.    30,635.         69.   3907. 
73.  26,884.         74.  53,994. 
78.  377,358.     79.  909,034. 
83.  $48,169.95. 
87.  $20,937.06. 
91.   5,965,009. 
2607.        96.    1492. 

100.    6898. 
104.  94,760,000  mi. 


70.   10,261. 
75.  751,470. 
80.  $37,685.17. 
84.  $38,062.68. 
88.  9,115,653. 
92.   2,791,654. 
97.   $9175. 

101.   6476. 
105.  $3693. 


Page  49.— 67.  11,483. 
71.   14,996.       72.  11,649. 
76.  386,669.     77.  151,488. 
81.  $29,081.46.      82.  $34,856.97. 
85.  $10,764.99.       86.  $25,209.82. 
89.   7,820,961.         90.  6,642,511. 
93.  $2710.        94.   $1890.         95. 

Page  50.  —  98.  $151.  99. 
102.  $  534,  gain.  103.  $  13,785. 
106.  $1895.  107.  $1010. 

Page  51.  — 108.  $215.  109.  $2390.  110.  $585.  111.  $738,  gain. 
112.  $1350.  113.  $4487.  114.  45,906.  115.  303,418.  116.  $9929. 

Page  58.  — 2.  1026.  3.  2064.  4.  1890.  5.  1708.  6.  1629. 
7.  3425.  8.  1137.  9.  2568.  10.  1540.  11.  3095.  12.  4224. 
13.  6324.  14.  6897.  15.  7176.  16.  26,936.  17.  35,526. 
18.  34,020.  19.  37,709.  20.  11,824.  21.  51,258.  22.  63,328. 
23.  31,066.  24.  39,925.  25.  59,898.  26.  62,818.  27.  103,136. 
28.  101,244.  29.  105,006.  30.  147,580.  31.  208,292.  32.  311,568. 
33.  622,242.  34.  284,235.  35.  311,632.  36.  375,201.  37.  273,875. 
38.  691,504.  39.  547,686.  40.  494,795.  41.  358,250.  42.  733,131. 
43.  738,944.  44.  $47.25.  45.  $112.50.  46.  $94.15.  47.  $130.80. 
48.  $137.55.  49.  $28.75. 

Page  59.— 50.  $224.70.        51.  $366. 
54.  28,800.      55.  $449.75.      56.  $338.40. 


52. 

57.  $62.80. 
62.  6144. 


53.  31,680. 
58.  $22.50. 
63.  $4025. 


59.  $460.25.      60.  $602.80.      61.  8869  Ib. 
64.   $42.75. 

Page  61.  - 1.  2740  ;  38,100  ;  9,314,000.    2.  3860  ;  61,000  ;  8,167,000. 


Page  62.  —  3.   4560;  90,300;  78,300,000. 


5.    3190;  31,000;  600,800,000. 
8.   7860  ;  244,200  ;  16,690,000. 
10.  11,360;  571,200;  20,416,000. 
12.  29,610  ;  785,600  ;  56,064,000. 
14.  37,740;  256.200;  60,718,000. 


4.   3750;  85,700; 

6.  4020;  41,600; 
9.  14,730;  147,800; 
11.  35,750;  396,500; 
13.  12,640;  95,400; 
15.  15,540;  506,700; 


51,690,000. 

678,500,000. 

24,368,000. 

20,514,000. 

53,217,000. 

48,828,000. 

Page  63.  —  20.  88,366.      21.  133,376.      22.  133,496.      23.  150,732. 

Page  64.  — 24.  148,311.  25.375,774.  26.201,880.  27.282,935. 
28.  311,872.  29.  206,886.  30.  146,069.  31.  233,988.  32.  340,405. 
33.  506,785.  34.  617,064.  35.  276,250.  36.  481,663.  37.  250,098. 
38.489,855.  39.619,344.  40.591,458.  41.11,234,275.  42.4,729,104. 
43.  19,022,724.  44.  1P,422,156.  45.  14,424,424.  46.  11,216,736. 
47.  23,103,018.  48.  13,977,718.  49.  14,896,404.  50.  9,765,217. 
51.  17,412,096.  52.  16,571,295.  53.  39,318,048.  54.  22,693,488. 


ANSWERS. 


441 


55.  221,347,750.  56.  305,221,392.  57.  2,026,441,428.  58.  3,841,167,050. 


61. 
64. 


59.  1,199,528,823.  60.   1,895,536,280. 

62.  $1,833,590.25.  63.   .$1,164,998.89. 

65.   $2,551,324.02.  66.   $1,533,115.75. 

68.   $3,831,397.66.  69.   $2,634,678.20. 

71.   $2,155,631.50.  72.   $2,528,601.90. 

74.    $1,234,742.52.  75.    $2,853,323.52.  76. 

77.    $1,254,133.50.        78.    $1,694,006.08.        79.    $2,842,474.80. 

Page  65.  —80.  $361.76.       81.  1140 mi.;  68,400  mi.       82.  1287  mi. 
83.   256  mi.  84.   $  101  gain.  85.    8073.  86.   417,872  Ib. 

87.    $470.60.        88.    134,096.       89.    $517.85. 

Page  66. —90.  $2.80.       91.  $495.       92.  $34.       93.  $11,834.73. 
94.    $61.20.        95.    $4350.        96.    545.        97.   $3587.86. 


2,361,055,599. 

$2,851,792.00. 
67.  $4,163,565.84. 
70.  $5,574,517.20. 
73.  $1,904,708.16. 

$2,111,073.36. 


Page  74. 

—  6.  906.    7.  427. 

8. 

1284. 

9.  667. 

10.  543. 

11.  1066. 

12.  507.   13.  1221. 

14. 

1502. 

15.  1216. 

16.  879. 

17.  1065. 

Page  75. 

—  18.  2646.   19.  603. 

2( 

>.  790. 

21.  1166. 

22.  2292. 

23.  1446. 

24.  729.   25.  947. 

26. 

1468. 

27.  490.   2 

18.  1387f. 

29.  603. 

30.  736.    31.  1704. 

32. 

662. 

33.  1496. 

34.  771. 

35.  1451. 

36.  923.   37.  1386. 

38. 

692  f. 

39.  1245. 

40.  405. 

41.  459. 

42.  307.   43.  451. 

44. 

1140. 

45.  1005. 

46.  656. 

47.  1058. 

48.  1587.    49.  661. 

50.  879. 

51.  901. 

52.  837. 

53.  793. 

54.  1171.    55.  522^ 

56.  6 

8,820i.    57. 

70,  335  1. 

58.  97,271|. 

59.  63,804.   60.  6* 

1,5984.   61.  $86.43.   6S 

\.  $38.45. 

63.  $73.47. 

64.  $85.40f.  65.  $8< 

5.14| 

.  66. 

$85.71|.   67. 

$44.17^. 

$117.044.          69.    $84.84|.  70.    $34.89|.  71.    $53.70f. 

72.  $122.45|.      73.  $75.69f,      74.  $43.75f.      75.  $1269f.       76.  145. 
77.    237.         78.    $294. 


Page  76.  —  79.    464. 
83.    1142.         84.    $1245. 
88.   $75.80.         89.    14,691. 

Page  77.— 4.   468T5^. 
8.    317.        9.    68T5^.        10. 
14.  54TV^.o 

fa. -II  agp1 

28. 


80.   $30.35. 
85.    $2186. 
90.  4087. 
5.  392  J&. 
31 


81. 


625. 
48. 


82.  $5.20. 
87.   $13.50. 


50. 
56. 
62. 
68. 

74.  9850; 
80.  9567. 
85.  55, 
89.  41,462 
93.  71,H4 


.  —  39.  313. 
45.  345. 
51.  762. 
57.  834. 
63.   785. 

69.  1347. 
75.  8764. 
81.  56,783. 
86.  ' 


40.  144. 

46.  435. 

52.  871. 
58.  872. 

64.  672. 

70.  2454. 
76.  7937. 


216. 


41. 

47.  326. 
53.  453. 
59.  625. 
65.  888. 
71.  3456. 
77.  7777. 

82.  53,232Mf.     83.  38,507fff. 
2,0594ff.         87. 
90.    60,433ih.        91. 
94.  73,471|H-         95- 


42.  384. 
48.  372. 
54.  875. 
60.  576A. 
66.  916. 
72.  5834. 
78.  5874. 


43.  613. 
49.  856. 
55.  644. 
61.  790. 
67.  1426. 
73.  6341. 
79.  7426. 
84.  59,632^. 
88.  6 1,002  |f|. 
92.  54,568|ff 
96. 


442  ANSWERS. 

97.  75,572ff£.  98.  73,439f£f.  99.  79,098ffg.  100.  50,422fff 
101.  68,816ffi.  102.  79,977|ff.  103.  82,786fff.  104.  72,672£f£. 

Page  81.  — 105.  30,416Ty¥V-  !06.  18,388^f.  107.  16,267ff£g. 
108.  18,383ffff.  109.  28,956ff£f.  110.  39,514|fff.  111.  61,410Sffi. 
112.  78,428ffff.  113.  .$36.  114.  217.  115.  342.  lie!  72. 
117.  3867.  118.  450.  119.  541.  120.  $  136|ff .  121.  2603^. 
122.  31.  123.  450. 

Page  82.— 124.  625.  125.  25.  126.  200  da.  127.  3520. 
128.  190.  129.  $4090.  130.  31.  131.  39,  and  420  sq.  miles 
over.  132.  5,  and  19,930  sq.  miles  over. 

Page  83.  — 3.  56.      4.  21.      5.  6.      6.  7.      7.  4.      8.  32.      9.  26. 

10.  27.     11.  2.     12.  21.     13.  115.     14.  321.     15.  270.     16.  6.     17.  31. 
Page  86.  — 2.    5.        3.    $110.50.        4.   $1.56.         5.   $134. 
Page87.— 6.  $1493.        7.    70.        8.    115.        9.   74.        10.  327. 

11.  96.         12.  $125.        13.  463.         14.  $11,640.        16.    $28,400. 
Page  88.— 17.  $13.75.      18.  $384.       19.  $88,  loss.      20.  264  mi. 

21.   549  mi.        22.   40.         23.    975.         24.    13.         25.   10,586  gal. 

Page  89.  — 26.  $3985.  27.  361^.  28.  7.  29.  3070.  30.  1704. 
31.  1383.  32.  180,604,125.  33.  3007.  34.  $626,  gain.  35.  10. 
36.  $3.50.  37.  $132. 

Page  90.  —  3S.  50.  39.  $33.41.  40.  $576.15.  41.  $5614.  42.  6. 
43.  $270,  horse;  $180,  buggy.  44.  116.  45.  $56.  46.  110. 
47.  510i|f. 

Page  95.  — 2.   32,  5.  3.  22,  3,  7.  4.  5«.         5.  2,  3,  5,  7. 

6.  32,  5,  7.       7.  24,  33.       8.  2,  3,  5,  11.       9.   22,  II2.       10.   23,  32,  5. 

11.  2,  3,  131.       12.  2*,  32,  13.       13.  2,  3,  5,  7,  11.       14.  32,  5,  7,  11. 
15.  32,  5,  72.      16.  22,  32,  52,  7.      17.  22,  3,  72,  13.      18.  2,  11,  13,  17. 
19.  2,  5,  7,  47.          20.  22,  3,  5,  7,  11.          21.  2«,  72.          22.  22,  953. 
23.  72,  11,  13.         24.  2,  3,  52,  29.         25.  2*,  72,  29.         26.  2i°,  3,  7. 
27.  2,  5,  7,  11,  13.          28.  2«,  5,  101.          29.  2™,  62.          30.  2?,  503. 
31.  2,  5,  7,  11,  41.      32.  22,  53,  97.       33.  2»,  3*. 

Page  96.— 2.  36.      3.48.      4.21.      5.  4i. 

Page  97.  — 6.  5|.      7.  13 J.       8.  6£.      9.  315,     10.  2.      11.  133. 

12.  24.       13.  10.       14.  105.      15.  90.      16.  25f      17.  444.      18.  84. 
19.  197J.        20.  192.        21.  31£.        22.  99.        23.  72.        24.  5440. 
25.  257i|.      26.  78£.      27.  5700.      28.  45.       29.  27. 

Page  98.— 30.  4J.  31.  40.  32.  $1.20.  33,  19.  34.  306. 
35.  50^.  36.  48.  37.  54.  38.  $6f.  39.  36. 

Page  103.- 2.  f§.  3.  ff.  *•  M-  5-  ft-  6-  **•  *•  tt- 

8.  |f.  9.  TW.  10.  T%V  11.  Jft.  12.  iff.  13.  iff.  14.  iff. 

15.  Hf.  16.  Ml-  17.  T<&.  18.  iff-  19-  W-  20.  tf|.  21.  flft- 

22-  m- 

Page  104.-2.  *;f.       3.  J;  f .      4.  § ;  J.       5.  f ;  J.      6.  f;f. 

7.  *;*.    8.  A;tt-    »-H;f    10-  H;  !•    "•  H;i-    12.  i;  f 


ANSWERS.  443 

13.  A;  A-    14.  H;  M-    15.  f;  §.    18.  f;  A-    17.f;f    18.f;f. 

19.  f ;  f.  20.  ft;  }f-  21.  |J  ;' A-  22.  £;  f.  23.  ft;  t- 
24.  if ;  if.        25.  f  ;  A-  26.  f ;  *.  27.  f ;  ft.  28.  jf ;  A- 
29.  f  ;  f .  30.  ^T  ;  |.  31.  ft  J  H-  32.  ft  ;  f  33.  f  ;  A- 
84.  f;A-        85.  AJtt-  36.  f ;  fa  87.  J;H-  38.  *5  A- 

39.  f;f.       40.  f?;Hf       «•  H  J  tt- 

Fagel06.  — 2.  if£.    8.  if*.    4.  Y<r«    5.  -5T2f.    &•  W-    7.  iff±. 

8.  -W--        9-  W-       10.  ^P.       11.  HI1-        12-  ^tf1-       13.  H**- 

14.  i-V^.       15.    A£fA.       16.  i^p.       17.   2  2^091.       18.  AJ^.1.       19. 

20.  ^-Vy--     81.  H8/--     22.  **^.     23.  ^p.    24.  A^fti.    25. 
26.  i|4;  l$±;  ^A;  ^  ;  ^J  Wi  W;  1I^- 

Page  107.— 2.  6TV;  2^.  3.  7J;31fr.  4.  7  ;  2||.  5.  5^;  3. 
6.  ^;  2fV  7.  4}-f;  Iff-.  8.  5| ;  3|f  9.  OJ ;  6^.  10.  6}f ;  6^. 
11.  6^J  5|f.  12.  6|f ;  634.  13.  20tf ;  15.  14.  12H;  10H- 

15.  lOff;  8|f.        16.  96if.        17.  90£f.         18.  93ff.         19.  77ff. 
20.  84.if.     21.  HOf.     22.  121.     23.  295&.     24.   343^.     25.  392TyF. 
26.  239.^TV     27.  170TW     28.  172|fJ.     29.  1892%V- 

Page^l08.-2.  T% .  ^.  A.    3.  f ;  f ;  f.    4.  T% ;  T\;  TV     5.  if; 

if ;  A-    6.  M  ;  i! ;  A-    7.  M ;  M ;  A-    8.  || ;  M ;  H-    9-  M ; 

A ;  H  -  i°-  M ;  A ;  A-  «•  M ;  f  * ;  &•  12.  If ;  A ;  A- 
13.  f ! ;  IE- ;  H-  14.  f f ;  H ;  ff.  15.  if ;  ^ ;  if.  16.  & ;  A  ; 

H-  i7-  if  5  It ;  M-  is-  A ;  A ;  A ;  A-  w-  A ;  M ;  A ;  A- 
20.  H;  it;  A;  H-  21.  M;  II;  fi;  if.  22.  «;  ft;  ft;  if. 

23.  if;  A;  If;  iS-         24.  |f;  ^;  if;  H-         25.  |f;  f  f ;  ft;  ff. 
Page  109. -27.    if;    1*5    H;    H-  28.   ||;    f  1 5   f*;    ft- 

29.  A ;  if ;  it ;  A-  30.  if  ;  ft ;  tf ;  fa  31.  f f ;  |f ;  ff ;  if. 
32.  if;  ff;  ff ;  if.  33.  f  | ;  ff;  ff;  H.  34.  Jfr ;  ^  J  Mf  J  iff- 
35.  if;  |f;  H;  |f  36.  f-|;  ff ;  ff  J  If-  37.  ff ;  ff ;  fj;  ff. 

38.  HI;  *H;  IH;  i^.    39.  ^;  ff ;  ff ;  ||.    40.  ^;  *;  T6A; 

i^V  «.  M;  M;  ^;  M-  42.  ff;  ||;  ff;  \\.  43.  Iff;  iff;  ftf; 
Mi-  44.  T^;  ^;  Afe;  iir-  45.  ^J  W?  ff;  ff-  46.  ^;  W; 

W;  W 

Page  112.  —  3.  -2ff.    4.  2f.    5.  1^.     6.2^.      7.  2ff.     8.  2ff. 

9.  2if.       10.  2J.       11.  2.       12.  Iff.       13.  2.       14.  If.       15.  1^. 

16.  2T\V   17.  lif.   18.  2f|.   19.  l|f.   20.  IJff.   21.  2ft. 
22.  2TV?.   23.  8ff.   24.  3^.   25.  2ff§.   26.  2A-   27.  l|ff. 
28.  2fi.   29.  22ff.   30.  22^.   31.  19^.   32.  22 fa   33.  17ff. 
34.  16T9¥.  35.  24|i.  36.  49i|.  37.  61^.  38.  147ff.  39.  101|^. 

40.  96iff.   41.  84f|. 

Page  113.— 42.  425|f  Ib.  43.  2610f f  bu.  44.  157ff  yd. 
45.  813T9^lb.  46.  $36J.  47.  $27311.  48.  20T\  yr.  49.  157T2I^mi. 
50.  17£f  yd. ;  f  68Jf.  51.  B,  31 J  ;  C,  44| ;  all,  89^  A.  52.  50ff£  A. 
53.  105^  mi. 

Page  115. -3.  fa      4.  if.       5.  A-       6.  ff.       7.  ff.       8.  A- 


444  ANSWERS. 

9.  f|.      10.  &.      11.  ^.      12.  /A-     13.  H-      14.  fW     15.  ^. 

16.  fV.      17.  flfc.      18.  ^.      19.  Iff.      20.  ft.     21.  rffr.      22.  |f. 

23.  fft.      24.  TW      25.  |£.      26.  T9?. 

Page  116. -27.  f£f.        28.  ft.        29.  A-       30.  flft.        81.  J. 

32.  TW       33.  1ft.        34.  2H-       35.  6ft.        36.  Stf.        37.  4T\V 

38.  2tfft.        39.  4|.         40.  2tf.         41.  5f£.        42.  1^.       43.  |ft. 
44.  IJf-         45.  H-         46.  3%.          47.  f|.         48.  if.        49.  Iff. 

50.  18ff.      51.  15|f       52.  10  ri^.       53.  4£.       54.  6rV       55.  6f|. 

56.  15f.         57.  10|.        58.  24^  yd.         59.  ffa  A.        60.  23f§  yd. 
61.  $.81}>      62.  Increased  f. 

Page  118. -2.  ft.      3.  i.      4.  1.      5.  |.      6.  f.      7.  ft. 

Page  119. -8.  f.  9.  f$.  10.  1J.  11.  ft.  12.  f  13.  ^. 
14.  ^.  15.  if  16-  23o-  17.  1.  18.  T|T.  19.  |.  20.  lrV 
21.  3.  22.  9.  23.  2i|.  24.  20|.  25.  4f.  26.  lii.  27.  1. 
28.  5|.  30.  189.  31.  375.  32.  488f.  33.  662f.  34.  1544. 
35.  770.  36.  1693rV  37.  1263.  38.  776|.  39.  998.  40.  1174£. 
41.  891f.  42.  1045^.  43.  460f.  44.  1059f. 

Page  120. —46.  95.   47.  129.   48.  413.   49.  539.   50.  488. 

51.  515.       52.  675.       53.  366.       54.  239i.       55.  510£.       56.  427f. 

57.  202f.       58.  433£.       59.  607f.       60.  87.       61.  199J.       62.  175. 

63.  238J.      64.  171f.      65.  170f.       66.  244.       67.  248.       68.  172f. 
69.  249§.     70.  248TV.     71.  273|f.     72.  790.     73.  507if.     74.  407 if. 

75.  337|.      76.  1026'j.      77.  582f.      78.  629.      79.  912.      80.  1432TV 
81.  14141.         82.  2937.         83.  1633T^7.         84.  634;|.         85.  1406. 
86.  1712|.          87.  2244|.          88.  $5225.          89.  $45.  90.  $25. 
91.  $27ii.      92.  f  116|f. 

Page  121.  — 93.  $61i.  94.  $2^.  95.  $105|.  96.  $16T\. 
97.  $121i.  98.  $62|.  99.  f  10,45"5.  100.  235|  Ib.  101.  $!!££. 
102.  508.Vmi.  103.  $129i.  104.  A,  $33^;  B,  $41 1.  105.  382^  ft. 
106.  $ 35].  107.  278|  mi. 

Page  123.  — 2.  f.  3.  |.  4.  ±%.  5.  1£.  6.  §.  7.  i|.  8.  |f 
9.  ff  10.  l^V-  11.  22V  12.  f .  13.  2§.  14.  f.'  15.  if!-  16.  H- 

17.  2f.     18.  171.     19.  i4|.     20.  10|.     21.  12f.     22.  10|.     23.  llf. 

24.  b\.    25.  8f.    26.  If.     27.  61.     28.  10.     29.  8.     30.  3i.     31.  2ff . 
32.  If      33.  2TV      34.  4.       35.  5*f       36.  4f.      37.  5r\.      38.  5J. 

39.  TV    40.  ^.    41.  _fv    42.  A.    43.  ^.    44.  ^.    45.  _2?. 

Page  124.—  46.  3^.    47.  f    48.  f.    49.  H-    50-  £•    51.  Iff. 

52.  T|T.       53.  J¥-       54.  1T\V      55.  2f      56.  ll^ft.      57. 

58.  llf|.      59.  &&.      60.  Ifff.      61.  ff|.       62.  7f7.      63. 

64.  2TW7¥.    65.  ¥%¥A-    66.  iffi-      67.  f|f|.     69.  £.     70. 
71-  aW^-      72.  Tff|¥.       73.  IT^flftV.      74.  jtfjfo.       75. 

76.  Iflff. 

Page  125.  — 78,  4f|.      79.  4ff.      80.  8|§.      81.  10f^.      82. 
83.  5||.         84.  5|.         85.  4^.         86.  4|§.         87.  4J|-        88.  3||. 


ANSWERS.  445 


89.  6|f.  90.  6|f.  91.  5TV  92.  8/?.  94.  7£.  95. 
96.  5rV  97.  6T\.  98.  10|f.  99.  7}f.  100.  6£f.  101.  5|f 
102.  6.  103.  2f|.  104.  8|f  .  105.  3TV  106.  7^  yd.  107.  11  bbl. 
108.  8}f  hr. 

Page  126.  —  109.  lOff  da.  110.  $1|£.  111.  10  children. 

112.  80  pairs.  113.  $4f.  114.  74fyd.  115.  147f|lb.  116.  25f|bu. 
117.  2112  steps.  118.  202i  |  A.  119.  63j*,  or  $  T%35.  120.  23i  bu. 
121.  $108Hf.  122.  $lljf.  123.  $5H- 

Page  127.  —  2.  ^f.  3.  $$.  4.  If  5.  1£.  6.  1TV  7.  10. 
8.  27.  9.  TV  10.  80.  11.  67£.  12.  ff.  13.  16^.  14.  3^T. 
15.  3T\V  16.  681.  17.  2H-  18.  2if.  19.  Itfjf.  20.  Jf  J|. 


Page  128.-  10.  &.         11.  if.         18.  A-         W.  ft.         14.  ^. 

15.  7\.  16.  A-  17.  Tfc.  19.  f  20.  ff.  21.  H.  22.  |f. 
23.  f.  24.  A-  25.  if.  26.  f  .  27.  ff.  28.  ff  29.  ||. 
30.  ff.  31.  20  times.  32.  10  times.  33.  llf  times.  34.  lOf  times. 
36.  f.  37.  ff.  38.  f.  39.  T\\.  40.  f  41.  ft-  42.  ^. 
43.  f|.  44.  ^¥.  45.  |i.  46.  T^.  47.  «•  48.  f.  49.  H- 
50.  ff.  51.  ff 

Page  129.—  2.  240.  3.240.  4.220.  5.216.  6.320.  7.280. 
8.  288.  9.  495.  10.  360.  11.  312.  12.  608.  13.  882.  14.  913. 
15.  1027.  16.  1065.  17.  600.  18.  1050.  19.  930.  20.  1044. 
21.  800.  22.  $810.  23.  $3360.  24.  $3237|.  25.  $132.30. 
26.  $34.64.  27.  2212  books. 

Page  130.  —  28.  2464  bu.     29.  $4308.    30.  1820  bu.    31.  $452.04. 

32.  $2288.     33.  $47,500.      34.5280ft.      35.  $42,000.     36.  Mr.  B., 
$56.      37.  $2322.      38.  $5080. 

Page  135.  —  1.  155TW    2.  $8|.    3.  llffhr. 

Page  136.—  4.  $14.58|.  5.  9065|.  6.  289f  A.  7.  33T}T  sq.  rd. 
8.  $8873.  9.  Increased  ¥\.  10.  Diminished  f.  11.  $46.  12.800 
Ib.  13.  iff.  14.  Tf  s.  15.  27£  gal.  16.  20ff  ^. 

Page  137.  -  17.  $  75|.  18.  A,  $  1704  ;  B,  $  1597|  ;  C,  $  968}. 

19.  $  Iff.  20.  24  bu.  21.  12|^.  22.  Iff  **•  &tff&r 

24.6.  25.  $44TV  26.  $6.25.  27.  3f  T. 

Page  138.  -28.  A,  T^  mi.    29.  $825.    30.  $19,3121.    31.  $1440. 

33.  $1^.     34.  Corn,  48;  wheat,  21|  ;  oats,  10|.     35.  $2.55|. 
Page  139.  -36.  $7466|.      37.  $  1662^.      38.  $8|.     39.  $1252; 

$2921^.  40.  $31TV  41.  $11^-  42.  36  da.  43.  $64.  44.  12f. 
45.  $76TV  46.  $.84^. 

Page  140.—  47.  A,  $6.20;  B,  $9.30.  48.  1st,  $10^;  2d,  $9T77. 
49.  B,  by  iahr.  50.  36  hr.  51.  $34|.  52.  A,  $4500;  B,  $5000. 
53.  8  children.  54.  R,  $  1260  ;  Q,  $  896  ;  W,  $  1204  ;  cost  of  drove, 
$3360.  55.  3rWff.  56.  $^f.  57.  $2000. 

Pagel41.-58.  $31,733£.     59.  $1558||.    60.  $1125.    61.  $286^- 


446  ANSWERS. 

62.  74}f  }  yd.      63.  $  14,875.      64.  The  1st,  ft  gal.  more  per  min. 
65.  Gain  $  56.       66.  2}f.       67.  $1520. 

Page   142.— 68.   1st,  $961fffi;    2d>   6768f|H;    3 
69.  1st,  $60#ft;  2d,  $18ftf$.    70.  $106|.     71.  5f  bu.    72. 
73.   Value   of  estate,  $  105,495  ;   elder  brother,  $  40,575  :  younger, 
$31,648};  sister,  $33,271}.      74.  ,$$*.      75.  62Ty/T  da.  ' 

Page  143.— 76.  16|  hr.  77.  $l}f.  78.  1st,  $35;  2d,  $40. 
79.  $45,937}.  80.  3^.  81.  A,  28  days  ;  B,  21  days.  82.  $9468. 
83.  9}  hr. 

Page  148. -2.  }.     3.  |.     4.  ft.     5.  ft.    6.  ft.    7.  ft.    8.  f.    9.  f. 

Page  149. -10.  jfo-  "•  AV  12.  AV  13.  ft-fr.  14.  /7. 
15.  ,flhr.  16.  flfo.  17.  ^.  18.  Iff.  19.  TVV  20.  Tfo. 

21.  ftftfo.      22.  ^AVo-     23.  S$MT.      24.  ftft.      25.  ^^.    27.  }. 
28.  T\.      29.  |.      30.  f$$.      31.  f.      32.  f|.       33.  ft$.       34.  f$}. 
35.  T\.      36.  jib,.       37.  T^e<y-      38.  f^ft.     39.  T^W-     40.  12TV^. 

41.  22*$.      42.  43|fi. 

Page  150.— 8.    .6.     9.   .25.     10.  .375.     11.  .8.     12.  .75.     13.  .15. 

14.  .56.       15.    .4.       16.    .333}.        17.    .555f.        18.   .24.       19.   .674. 
20.  .923^.      21.    .533}.     22.  .2916|.     23.  1.5.     24.    .055|.     25.   .12. 

26.  .1875.       27.  .735^.    28.    .545T\.     29.   .204.      30.  .056.     31.  .35. 

32.  .45.       33.    .16666  +  .        34.    .25.       35.    .42857  +  .       36.    .8169+. 

37.  .98666+.       38.   1.45454+ .       39.    12.5.       40.    18.75.      41.    24.6. 

42.  .25.       43.    .04761  +  .       44.   37.5.       45.    .4525.       46.   16.4444+. 
47.  48.53.    48.  .23625.    49.  60.08.     50.  .0001733+.     51.  513.00666+. 
52.  75.0005. 

Page   151.— 2.    52.234.         3.  95.7953.        4.  42.2717.         5.  9.88. 

6.  9.5824.       7.  54.4174.        8.  138.8875.       9.  109.185.       10.  294.534. 

11.  247.59.        12.    126.7814.          13.    535.12112.         14.    260.6319605. 

15.  $234.15.         16.   $343.10.         17.    $205.56^.          18.   $2020.78|. 
19.  2.775  tons. 

Page  152.  — 20.  $38.50.  21.  39.385  cords.  22.  120115.054048. 
23.  $102.50.  24.2301.9779.  25.  $11.58.  26.65.09895.  27.  $32,653.21. 
28.  $11,218.03. 

Page  153.—  2.  .219.     3.  .1682.     4.2.3038.     5.2.2215.     6.12.2124. 

7.  305.09746.     8.  16.763807.     9.  106.1226.     10.  18.332.     11.  11.079. 

12.  230.25.     13.  158.846.      14.  665.8794.     15.  646.63.      16.  999.999. 
17.  $25.915.     18.  $27.17.      19.  $35.67.      20.  $25.60.      21.  $86.25. 

22.  $58.25.     23.  $9.96.    24.  $99.62}.     25.  $181.84}.    26.  $60.22}. 

27.  $221.59}.     28.  1433.525  T. 

Page  154.  — 29.  $58.88.    30.  484.006651.     31.  $2.17.    32.  $10.13. 

33.  $2566.74.  34.  147.875  bbl.   35.  .621.   36.  $4.10J.    37.  $274,741.89. 

38.  $3,396.03. 

Page  155.  —  2.   .08.     3.2.304.    4.4.410.     5.   .2795. 

Page  156.  — 6.  160.08.  7.  .23328.  8.  25.752.  9.  1.152.  10.  .4. 
11.  3.8.  12.  10.692.  13.  .1785.  14.  58.05.  15.  386.4.  16.  74.375. 
17.  .024288.  18.  1199.  19.  21.528.  20.  29.495.  21.  .3136. 

28.  46.875.     23.  197.775.     24.  2.25225.     25.  151.75836.    26.  272.80767. 
27.  393.225.      28.  27.      29.  .825.      30.  2.048.     31.  .7272.      32.  1.16. 


ANSWERS.  447 

33.  24.869.  34.  1714.0102.  35.  76.327.  36.  10,135.97504. 

37.  .00001032.  38.  5.251.  39.  429.66.  40.  30,574.45.  41.  1331.34945. 
42.  732.09636.  43.  .0063612.  45.  583.6.  46.  1683.4.  47.  95,817. 
48.  373,186.  49.  77.12.  50.  148.11.  51.  2963.5.  52.  35,682. 
53.  128,509.2.  54.  3,505,692.  55.  4,579,880.  56.  4880.556. 

Page  157.  — 57.  $30.8U.  58.  $22.343f.  59.  $8238.75.  60.  818.75 
yd.  61.  13,599.1  tons.  62.  $19.295.  63.  $3.26£.  64.  $805.50. 
65.  $27.595.  66.  $31.155. 

Page  159. —2.  1.39.  3.  21.5434+.  4.  .25.  5.  .25.  6.  42.8. 
7.  .0005.  8.  16.89513+.  9.  87.5.  10.  .00365.  11.  .763. 
12.  356. 11111 +  .  13.  30.2.  14.  2.13.  15.  12.24.  16.  1485.60159  +  . 
17.  .15.  18.  3650.  19.  73.21.  20.  27,500.  21.  .475.  22.  2643.6923+. 
23.  .21.  24.  .916.  25.  12310.7.  26.  .027.  27.  790.  28.  .0066. 
29.  .01.  30.  100.  31.  .00005.  32.  6000.  33.  360.  34.  7580. 
35.  .0000561. 

Page  160. —37.  3.3.  38.2.4125.  39.1.91238.  40.2.4187. 
41.  .0993405.  42.  .01529.  43.  .0023719.  44.  5.322.  45.  711.25. 
46.  .072345.  47.  2.31206.  48.  .00017216.  49.  .1073035.  50.  .206288. 
51.  $12.  52.  5.557+ T.  53.  17  harrows.  54.  45  bbl.  55.  128 
doz.  56.  $8.50.  57.  88 }  bbl.  58.  50  bu.  59.  27  bureaus.  60.  9  T. 

Page  161.—  2.  3,947,625.    3.2,382,999.   4.1,451,870.    5.42,117,840. 

6.  74,703,294.  7.  40,844,133.  8.  34,439,944.  9.  50,320,032.  10.  672,544,296. 

11.  860,796,984. 

2.  215,334.      3.  373,116.      4.  1,585,567.      5.  493,416.     6.  1,670,559. 

7.  1,620,402.     8.    723,114.        9.  2,190,832. 

Page  162.  — 10.  837,408.  11.2,149,888.  12.4,323,228.  13.8,049,776. 

2.  95,600.       3.  210,900.        4.   114,900.        5.  224,700.        6.  146,600. 

7.  107,400.      8.  81,550.      9.  538,200.      10.  2,448,500.      11.  2,366,300. 

12.  321,000.     13.  431,600.     14.  515,500.     15.  798,000.     16.  315,000. 
Page  163.— 2.  $26.73|.     3.  $78.421.     4.  $5.77$.     5.  $69.60625. 

6.  $45.93|.     7.  $164.62$.     8.  $5.97.      9.  $89.99375.      10.  $84.81^. 
11.  $32.721.     12.  $278.73.     13.  $  47.81  J.     14.  $52.32. 
Page  165.  — 2.  $6.23.         3.  $44.90.         4.  $23.35.         5.  $4.36. 

6.  $74.25.       7.  $94.855.       8.  $115.50.       9.  $76.525. 

Page   166.— 10.    $130.075.  11.    $34.50.  12.    $595.27$. 

13.  $574.86f.       14.  $446.93.       15.  $37.70.       16.  $76.95|. 
1.  21  Ib.     2.  26  shovels.     3.  58,317.798  gr. 

Page  167.— 4.  150  bu.  5.  $10.626.  6.  $19.76£.  7.  21,504,200 
cu.  in.  8.  A,  135  A.;  B,  225  A.  9.  200  bu.  10.  .00144. 
11.1,000,000.  12.  .001.  13.  18  da.  14.  148||  bu.  15.  315.75  mi. 
16.  $87.  17.  $2268.895.  18.  $11,377. 

Page  168.  — 19.  578.31  +  .  20.  .33075.  21.  5372.136  ft.  22.  139.91 
mi.  23.  .17899+.  24.  $39.25.  25.  $75.  26.  $19.86|,  27.  $3598.56. 
28.  1045.769  A.  29.  $4.50  gain. 

Page  169.— 30.  102.722.  31.  $5.375+.  32.  .677857  +  . 

33.  .05903  +  .  34.  69i  C. ;  $293.78|.  35.  $14,061.921  +  .  36.  $195.85 
gain.  37.  .053£.  38.  $110.10.  39.  $30.52$. 

Page  172.  —  2.  74  ft.     3.  109  ft.     4.  96 $  ft.     5.  231}  ft.     6.  501  ft. 

7.  10,815$  ft.      8.  16,264$  ft.      9.  28,065  ft.     10.  98  in.     11.  124  in. 


448  ANSWERS. 

12.  210  in.     13.  5098  in.     14.  198,103  in.     15.  326,920  in.     17.2yd. 
7t  in.     18.  2  yd.  21  in.     19.  3  yd.  2  ft.  Of-  in.     20.  4  yd.  2  ft.  101  in. 
2i.    106  rd.  3  yd.  2  ft.      22.  137  rd.  2  ft.  4f  in.      23.  177  rd.  4  yd.  10 
in.     24.  186  rd.  3  yd.  2  ft. 

Page  173.  —  26.  2  ft.  3  in.  27.  2  ft.  10^  in.  28.  1  ft.  10}  in. 
29.  2  ft.  7*  in.  30.  2  yd.  2£  in.  31.  8  yd.  1  ft.  7.71  in.  32.  232  rd. 
33.  312  rd. 

Page  174.  —2.  6  rd.  3  yd.  2  ft.  3.  21  rd.  2  yd.  2  ft.  4.  1  mi. 
138  rd.  1  yd.  5.  9  mi.  190  rd.  5  yd.  6.  24  mi.  58  rd.  1  yd.  7.  1  mi. 
128  rd.  2  yd.  1  ft.  4  in.  8.  1  mi.  167  rd.  2  ft.  9.  14  mi.  106  rd.  3  yd. 
2  ft.  10.  55  mi.  123  rd.  3  yd.  1  ft.  6  in.  11.  18  mi.  225  rd.  3  yd.  2 
ft.  6  in.  12.  56  mi.  261  rd.  3  yd.  1  ft.  6  in.  13.  6  mi.  229  rd.  3  yd. 
2  ft.  10  in.  14.  9  mi.  80  rd.  1  ft.  8  in.  15.  14  mi.  176  rd.  5  yd.  1  ft. 
16.  13  mi.  266  rd.  1  yd.  1  ft.  4  in.  18.  ^  rd.  19.  Tf  ^  rd.  20.  ,%  rd- 
21.  2hrd-  22.  m>76?  rd-  23.  -sl-g  rd-  24.  ^  mi.  25.  „*„  mi. 
26.  j^v  mi.  27.  T¥^o  mi-  28.  ^fav  mi.  29.  y^W  mi- 

Page  175.  —  31.  .7070707+  rd.  32.  .81818+  rd.  33.  .510101  + 
rd.  34.  4.631313+  rd.  35.  8.3484848+  rd. 

Page  177.  —  1.  5904  sq.  in.  2.  12,096  sq.  in.  3.  26,496  sq.  in. 
4.  318,816  sq.  in.  5.  471,456  sq.  in.  6.  4,704,600  sq.  in.  7.  31,389,120 
sq.  in.  8.  51,945,300  sq.  in.  9.  188,375,220  sq.  in.  10.  8,032,115,520  sq. 
in.  11.  5  sq.  yd.  6  sq.  ft.  116  sq.  in.  12.  5  sq.  yd.  1  sq.  ft.  96  sq.  in. 

13.  6  sq.  yd.  1  sq.  ft.  80  sq.  in.       14.  45  sq.  rd.  9  sq.  yd,  7  sq.  ft.  108 
sq.  in.       15.  1  A.  90  sq.  rd.  17  sq.  yd.  4  sq.  ft.  72  sq.  in.        16.  1  A. 
159  sq.  rd.  28  sq.  yd.  2  sq.  ft.  36  sq.  in.      17.  1  sq.  mi.      18.  3  sq.  mi. 
503  A.  10  sq.  rd.     19.  91  A.  16  sq.  rd.  18  sq.  yd.  1  sq.  ft.     20.  18  sq. 
yd.  8  sq.  ft.  22£  sq.  in.       21.  100  sq.  rd.       22.  25  sq.  yd.  8  sq.  ft.  51^ 
sq.  in.       23.  16  sq.  yd.  4  sq.  ft.  54.18  sq.  in.       24.  5  sq.  ft.  36  sq.  in. 
25.  126  sq.  in.      26.  7  sq.  yd.  2  sq.  ft.  48  |f  sq.  in.      27.  79  sq.  rd.  6 

.  rd. 


sq.  yd.  64*  sq.  in.  29.  ^f  7  sq.  rd.  30.  T^T  sq.  yd.  31.  rffa  A. 
32.  .632716+  sq.  yd. 

Page  179.  —  1.  26,040  cu.  in.  2.  55,410  cu.  in.  3.  1,897,344  cu.  in. 
4.  2,842,560  cu.  in.  5.  3,760,128  cu.  in.  6.  456,192  cu.  in.  7.  1,128,384 
cu.  in.  8.  27,035  cu.  in.  9.  3  cu.  yd.  5  cu.  ft.  152  cu.  in.  10.  2  cu. 
yd.  2  cu.  ft.  12  cu.  in.  11.  1  cu.  yd.  13  cu.  ft.  755  cu.  in.  12.  15  cu. 
yd.  14  cu.  ft.  538  cu.  in.  13.  9  cu.  yd.  3  cu.  ft.  1702  cu.  in.  14.  129 
cu.  yd.  11  cu.  ft.  228  cu.  in.  15.  23  cu.  ft.  1080  cu.  in.  16.  1440  cu. 
in.  17.  20  cu.  ft.  432  cu.  in.  18.  1166.4  cu.  in.  19.  A-  cu.  yd. 
20.  .30144  cu.  yd.  21.  $17.65. 

Page  180.  —  1.  10151.     2.  87  H.     3.  1401.    4.  7641.     5.  25951. 

6.  4060  1.      7.  7575  1.     8.  97,620  f.     9.  1  rd.  12  1.  3.96  in.     10.  57  ch. 
14  1.  2.26  in.      11.  15  1.      12.  3  rd.  5  1.      13.  54  ch.  14.  2  rd.  9  1.  3.96 
in.     15.  1  ch.     16.  8  ch.  3  rd.  1  1.     17.  76  ch.  3  rd.  10  1.     18.  1  mi.  4 
ch.  1  rd.  11  1.      19.  1  mi.  16  ch.  2  rd.  7  1.  5.56  in.      20.  1  mi.  75  ch.  3 
rd.  12  1.  6.96  in.    21.  ^  mi.     22.  71  \  1.     23.  ^  ch. 

Page  181.  —  1.  50,000  sq.  1.  2.  448  sq.  rd.  3.  2  rd.  12  1.  3.96  in. 
4.  24,150,000  sq.  1.  5.  310,320  sq.  rd.  6.  6  sq.  ch.  7  sq.  rd.  125  sq.  1. 

7.  89  A.      8.  6  A.     9.  3  sq.  rd.      10.  192  A.      11.  3  sq.  rd.  555  sq.  1. 
12.  300  A.         13.  5  sq.  rd.  250  sq.  1.         14.  46  A.  8  sq.  ch.  12  sq.  rd. 
15.  13  sq.  rd.  515  sq.  1. 


ANSWERS.  449 

1.  828  gi.        2.  884  gi.          3.  986  gi.        4.  1  qt.  1  pt.         5.  510  gi. 

6.  1463  gi.       7.  1  qt.  1  pt.       8.  2  qt.  1  pt.  2f  gi. 

Page  182.  — 9.  24  gal.  2  qt.  10.  13  gal.  1  qt.  1  pt.  11.  5  bbl. 
12  gal.  2  qt.  12.  1  bbl.  30  gal.  1  qt.  1  pt.  13.  10  bbl.  6  gal.  2  qt. 

14.  6  bbl.  17  gal.     15.  14  bbl.  15  gal.  3  qt.  1  pt.     16.  61  bbl.  4  gal.  2  qt. 
17.  25  bbl.  12  gal.  2  qt.     18.  J-0  gal.     19.  ^  gal.     20.  .1875  gal. 

1.  61  pt.  2.  91  pt.  3.  314  pt.  4.  38*  pt.  5.  187  pt.  6.  423  pt. 

7.  967 £  pt.  8.  10  pt.  9.  67  bu.  10.  105  bu.  11.  171  bu.  3  pk.4  qt. 
12.  72  bu.  2  pk.  13.  58  bu.  3  pk.  14.  150  bu.  15.  3  bbl.  28  gal.  1  qt. 
2  gi.   16.  122  bu.  3  pk.  7  qt.   17.  7  bbl.  29  gal.  2  qt.   18.  ^f  <j  bu. 
19!  2$ T  bu.  20.  .96875  bu. 

Page  183. —1.  6812  oz.  2.  6024  oz.  3.  3497  oz.  4.  12,000  oz. 
5.  11,460  oz.  6.  120,130  oz.  7.  176,392  oz.  8.  67  Ib.  8  oz.  9.  87  Ib. 
8  oz.  10.  66  Ib.  11.  1  cwt.  30  Ib.  12.  2  T.  2  cwt.  60  Ib.  13.  3  T. 
15  cwt.  25  Ib.  14.  4  T.  1  cwt.  23  Ib.  15.  9  cwt.  24  Ib.  16.  18  T.  4 
cwt.  50  Ib.  17.  30  T.  17  cwt.  31  Ib.  18.  52W  cwt.  19.  ^Vff  T. 
20.  .00275  T.  21.  W^T. 

Page  184.  —  1.  1892  gr.  2.  3658  gr.  3.  5196  gr.  4.  7  oz.  4  pwt. 
5.  50,778  gr.  6.  262,887  gr.  7.  10  oz.  10  pwt.  8.  14  pwt.  Of-  gr. 
9.  4  oz.  20  gr.  10.  5  oz.  6  pwt.  16  gr.  11.  5  Ib.  3  oz.  16  pwt. 
12.  18  Ib.  7  oz.  13.  2  Ib.  6  oz.  5  pwt.  14.  102  Ib.  9  oz.  16  pwt. 

15.  658  Ib.     16.  40  Ib.  3  oz.  18  pwt.    17.  6  Ib.    18.  ^  Ib. 
19-  ireVooO2-    20.  .27083  +  Ib. 

Page  185.  —  1.  475  gr.  2.  3220  gr.  3.  89,000  gr.  4.  141,255  gr. 
5.  12  Ib.  10  oz.  7  dr.  6.  14  Ib.  9  oz.  4  dr.  7.  38  Ib.  8.  28  Ib.  8  oz. 

1  dr.  1  so.     9.  2  Ib.  2  oz.  1  dr.  1  sc.     10.  16  Ib.  3  dr.  1  sc.  15  gr. 
1.  $32.91$.     2.  $  26.66 ;{.     3.  ^f.    4.  $2.12T4T. 

Page  186.  — 1.  18,912  sec.      2.   23,258.       3.   469   hr.      4.   17  hr. 

8  min.  34f  sec.     5.  720  hr.     6.  348  hr.     7.  994  hr.     8.  18  hr.  50  min. 

24  sec.     9.  1  da.  20  min,      10.  21  wk.  5  da.      11.  44  wk.  1  da.  16  hr. 

12.  47  wk.  4  da.  8  hr.          13.  71  wk.  3  da.          14.  6  wk.  3  da.  11  hr. 

15.  5  da.  15  lir.  15  min.  50  sec.        16.  1  wk.  3  da.  1  hr.  4  min.  56  sec. 

17.  1  wk.  6  da.  8  hr.  30  min.       18.  ^Vw  da- 
Page  187. —4.  123,163".  5.  130°  9' 20".  6.  IS.  24°  20'. 

7.  128,478";  67,034".      8.  136°.       9.  105°  56' 40".       10.  68°  58'  20". 

11.  88°  43' 20".     12.  135°  5'.     13.  108°  13' 20". 

1.  2424  far.     2.  12,743  far.     3.  13,749  far.     4.  19,803  far.    5.  33,921 

far.     6.  43,383  far.     7.  72,000  far.     8.  7s.  Qd.     9.  13s.     10.  9s. 
Page  188.— 11.  £36  Id.  3  far.       12.  £89  18s.  10<2.       13.  £4  Id 

2  far.     14.  £  37  9s.  4d.     15.  £68  15s.     16.  £  197  8s.  Sd.    17.  £50  12s. 
6d.          18.  £145  4s.  Sd.         19.  £  50.         20.  £.175.        21.  £  .283£. 
22.  £3.3208  +  .       23.  £5.4155.       25.  $152.422  +  .       26.    $118.803+. 
27.  $144.90  +  .      28.  $249.651  +  .      29.  $172.436  +  .      30.  $89.868  +  . 
31.    $50.226  +  .       32.    $172.314+.       33.   $74.781  +  .       34.    £81  4s. 
3.9+  far.      35.  £50  14s.  9d.  3.1+  far.     36.  £117  15s.  lOd.  3.3+  far. 
37     £126  Us.  .8+  far.     38.  £48  14s.  Id.  3.8+  far.     39.  £87  14s.  Id. 
3.1  +  far.    40.  £66  12s.  6d.  3.8+  far.     41.  £205  9s.  Sd.  3+  far. 

Page  189.  —2.  83  Ib.  10  oz.  3  pwt.  3  gr.     3.  186  bu.  2  pk.  4  qt.  1  pt. 
4.  80  T.  17  cwt.  6  Ib.  5  oz. 
Page  190.  —  5.   89  gal.  1  pt.  1  gi.         6.    126  Ib.  7  5  6  3  2  3  8  gr. 


450  ANSWERS. 

7.  92  mi.  162  rd.  1  yd.  2  ft.  8  in.  8.  145  sq.  rd.  25  sq.  yd.  2  sq.  ft.  58 
sq.  in.  9.  £320  13s.  lOd.  1  far.  10.  1222  cu.  yd.  2  cu.  ft.  149  cu.  in. 
11.  76°  23'.  13.  23  wk.  6  da.  8  hr.  42  min.  40  sec.  14.  £9  13s. 
15.  13  cwt.  78  Ib.  8|  oz. 

Page  191.  — 2.  3  Ib.  10  oz.  19  pwt.  3  gr.  3.  £4  15s.  2d.  4.  1  Ib. 
8  5  2  3  1  3  5  gr.  5.  22  gal.  3  qt.  1  pt.  6.  1  mi.  306  rd.  10  in.  7.  9 
gal.  1  qt.  1  pt.  2  gi.  8.  3  hr.  40  min. 

Page  192.  — 10.  2s.  Id.  I  far.  11.  21  rd.  3  yd.  5.3  in.  12.  5  da. 
20  hr.  36  min.  13.  14  bu.  2  pk.  5.232  qt.  14.  167  rd.  2  ft.  6  in. 
15.  45°  49'  15".  17.  32  yr.  2  mo.  15  da.  19.  5  yr.  5  mo.  28  da. 

20.  36  yr.  16  da.  old. 

Page  193.  —2.  102  bu.  2  pk.  4  qt.  3.  115  Ib.  11  oz.  16  pwt.  6  gr. 
4.  £  22  2s.  Qd.  5.  37  cwt.  26  Ib.  12  oz.  6.  389  gal.  1  qt.  1  pt.  7.  35 
bu.  2  pk.  7  qt.  8.  7304  da.  20  hr.  16  min.  34  sec.  9.  3  Ib.  6  oz.  7  pwt. 
12  gr.  10.  404  bu.  3  pk.  5  qt.  11.  103  cu.  yd.  25  cu.  ft.  432  cu.  in. 

Page  194.  —2.  11  Ib.  9  oz.  15  pwt.  18  gr.  3.  16  T.  2  cwt.  38  Ib. 
4.  13  hhd.  3  gal.  2  qt.  1  pt.  3£  gi.  5.  13  mi.  319  rd.  2  yd.  1  ft.  6.  10 
C.  39  cu.  ft.  1100  cu.  in.  7.  2  Ib.  7  oz.  15  pwt.  17 1  gr.  8.  7  rd.  1  yd. 
2  ft.  9.  12  mi.  139  rd.  10.  9  cwt.  42  Ib.  11.  2°  14'  27". 

Page  195.— 13.  7.  14.6.  15.15.083  +  .  16.  25  da.  17.  20  sacks. 

18.  9  bales.   19.  78  hr.  18  min.  33.2+ sec.   20.  $  60.85.   21.  160 
pickets.  22.  187  medals.  23.  160  ft. 

Page  196.  — 1.  $.83f.  2.  $2.28.  3.  93 boxes.  4.  $3.70.  5.  7fbu. 
6.  500  sq.  yd.  7.30ft.  8.41,775,360ft.  9.5ft.  10.  271.32  rd. 
11.  47,322  in.  12.  80  rd.  13.  90  sq.  ft.  14.  $35.20.  15.  $54.88. 

Page  197.  —16.  48  bars.   17.  4281 /T.   18.  66  yr.  10  mo.  29  da. 

19.  35  C.  4  cu.  ft.      20.  179  A.  130  sq.  rd.  2  sq.  yd.  5  sq.  ft.  36  sq.  in. 

21.  6f  qt.        22.  rfa  Ib.       23.  43  sq.  rd.  19  sq.  yd.  2  sq.  ft.  36  sq.  in. 
24.  792.      25.  .01015625  bu.      26.  84.       27.  27  A  bbl-       28-  *f  *  yr- 
29.  U.     30.  .17708+ da.    31.  .005765625  gal.    32.  $.    33.  2  oz.  5  pwt. 
34.  102  rd.  4  yd.  1  ft.  8  in. 

Page  198.  —  35.  $  188.73.  36.  47  cups.  37.  53 J  bags.  38.  Lead, 
1240  gr.  39.  Silver,  42£  gr.  40.  139  Ib.  9  oz.  1  pwt.  16  gr.  41.  1760 
rails.  42.  35¥5T  mi.  43.  $19,406.25.  44.  280  rd.  2yd.  4  in.  45.  ?f  fo  mi. 
46.  $282.135.  47.  25 bu.  3pk.  4qt.  1.92+  pt.  48.  $15.739+.  49.  $2680. 

Page  199.— 50.  56U  mi.  51.  34.858+  cu.  ft.  52.  3°  17' 55". 
153.  76°  20' 25".  54.  $259. 87 J.  55.  18  bu.  2  pk.  7£  qt.  56.30.03  + 
ch.  57.  16|da.  58.  $20.85.  59.  4.447+  bbl.  60.  $1.43+.  61.  34 
spoons.  62.  13,040;  $74.98. 

Page  200.— 63.  $.39 1.  64.  $193.  65.  $14.17^.  66.  $2050.06. 
67.  $12.87.  68.  20  powders.  69.  92  pills.  70.  $.83f|.  71.4.528+ 
bbl.  72.  $54.85.  73.  140  coins.  74.  49  bags. 

Page  202. — 2.  3  hr.  5  min.  2  sec.     3.  5  hr.  8  min.  11}  sec. 

Page  203.  —4.  54  min.  31f  sec.  5.  50  min.  55  sec.  after  8  o'clock. 
6.  23  min.  57£  sec.  before  twelve.  7.  1  hr.  17  min.  20T\  sec.  8.  6  hr. 
31  min.  18}f  sec.  9.  9  min.  21^  sec.  slow.  10.  53  min.  50  sec.  after 
noon  Jan.  1. 

Page  204.  —  2.  49°  5'  45".  3.  19°  3'  30".  4.  15°  35'.  5.  22°  30'. 
6.  10°  53'.  7.  73°  54'  25".  8.  122°  24'  15"  west.  9.  76°  50'  west 
10.  East  23°  45'.  11.  2°  20'  east. 


ANSWERS.  451 

Page  206.  —  1.    1  A.  20  sq.  rd.      2.    5  A.  64  sq.  rd.      3.    180  sq.  yd. 

4.  3555f  sq.  yd.     5.    31 1£  sq.  yd.    6.    10  sq.  ft.  72^  sq.  in.     7.    8294.4 
sq.  ft.     8.    4319.84  sq.  mi.     9.    2236  sq.  ft.     10.    X  A.     11.    $308.25. 
12.  40  A.     13.   120  A.     14.  $11,900.     15.  24yd.     16.  40  rd.    17.  6ft. 
18.  60yd.     19.  200  rd. 

Page  207.  — 1.  460  sq.  ft.  2.  624  sq.  ft.  3.  705  sq.  ft.  4.  31,535 
)  sq.ft.  5.  48,500  sq.  ft.  6.  14,437$  sq.  ft.  7.  2970  sq.  yd.  8.46,750 
sq.  rd.  9.  22ft.  10.  92ft.  11.  13$  rd.  12.  7  rd.  13.  160  rd. 

Page  208.  —  1.  180 sq.  ft.     2.  675 sq.ft.    3.  518 sq.ft.    4.  1161  sq.ft. 

5.  1842$  sq.ft.     6.  500 sq.ft.     7.  7$|$A.     8.  $677.30+.     9.  364sq.ft. 
Page  209.  —  1.   130  sq.  ft.      2.   180  sq.  ft.      3.  342  sq.  ft.     4.637$ 

sq.  ft.      5.  2139  sq.  ft.      6.  9£  A.      7.   13|  A.     8.  336  A.     9.  114f  A. 

10.  82$  A.     11.  80  sq.  ft. 

Page  210.  — 1.  110ft.  2.  242ft.  3.  198ft.  4.  154ft.  5.  264ft. 
6.176ft.  7.  308ft.  8.  330ft.  9.  396ft.  10.  528ft.  11.  47.124ft. 
12.62.832ft.  13.75.3984ft.  14.59.6904ft.  15.91.1064ft.  16.109.956 
ft.  17.  50.2656  ft.  18.  78.54  ft.  19.  97.3896  ft.  20.  141.372  ft. 

Page  211.— 22.  4.5518+  ft.  23.  52.2026+  ft.  24.  101.2223+ 
ft.  25.  135.5996+  ft.  26.  213.2671+  ft.  27.  304.1443+  rd. 
28.  381.9709+ rd.  29.  533.4861+  rd. 

1.  198.937|  sq.  ft.  2.  286.47  sq.  ft.  3.  113.0976  sq.  ft.  4.  7854 
sq.  ft.  5.  32,592  sq.  ft.  6.  60093.66+  sq.  rd.  7.  42174.68+  sq.  rd. 

8.  45239.04  sq.  rd.         9.   12271.875  sq.  ft.          10.  104062.3584  sq.  rd. 

11.  8148.48  sq.  rd.       12.   94.5457+  A. 

Page  212.— 1.    $42.93.       2.    $19.67$.       3.   $27.77.      4.  $21.68. 
5.  $23.84.       6.  $8.82$.       7.  $57.545. 
Page  213.— 1.    $61.20.       2.    $72.       3.    48yd.      4.    $84;  $88.08. 

5.  $48;  $50.40.       6.  49fyd.;  $91.76.       7.  $64.59. 

Page  214.  —  8.    $22.125.  1.   14  rolls.  2.    7  double  rolls. 

3.  8  double  rolls,  1  single  roll. 

Page  215.  — 4.    8  double  rolls.  5.    7  double  rolls,  1  single  roll. 

6.  $24.40. 

1.  $50.       2.  $29.70.       3.   $44.       4.   $117. 

Page  216. —1.    2640  bricks.          2.    $62.524.          3.    51  $  perches. 

4.  $86.88+.       5.  $708.48.       6.  $830.39. 

1.  3f  C.       2.  3J|  C.       3.  3^  C.       4.  4^  C. 
i)    Page  217.— 5.  $4.43.      6.  $14.84|.       7.  $14.355+.      8.  12||  C. 

9.  60  C. 

1.  15ft.       2.   17$  ft.       3.  24ft.       4.  15ft. 

Page  218.— 5.   150ft.     6.  196ft.     7.  162ft.     8.  $35.     9.  $21.60. 

10.  $1.18f. 

1.  12.053+  bu.     2.  24.106+  bu.     3.  52.231+  bu.     4.  84.374+  bu. 

5.  4.977+  ft.      6.   77.343+  bu. 

1.  14  bbl.  7.8  gal.     2.  13  bbl.  9.4  gal. 

Page  219.— 3.    24  bbl.  18.2337+  gal.  4.    49  bbl.  12.448  gal. 

5.  1496.1038+  gal.       6.  53.716+  bbl.      7.  190.99+  bbl. 

1.   48  bu.          2.    70  gal.        3.   68$  bbl.        4.    6|  T.        5.   6T\  T. 

6.  16M  T.;   16$f  T.;  16  T. 

Page  224.  — 1.  $158.33$.  2.  14  da.  3.  $213.  4.  665  mi. 
5.  3360  bu.  6.  60  A.  7.  $83.16.  8.  10  men. 


452  ANSWERS. 

Page  225.  — 9.  $13.32.  10.  $6461.53.  11.  1081  A.  12.  $162.71. 
13.  52  wk.  14.  14  kegs.  15.  $68.57.  16.  $50,575;  $57,800. 
17.  $97.849+.  18.  51|  Ib.  19.  22 A  T.  20.  90f  A.  21.  E,  13,910 
A.;  F,  12,840  A.  22.  A,  40,327  Ib.;  B,  51,849  Ibi ;  C,  5761  Ib. 

Page  226.  —  23.  44021 ;  4829£.  24.  /&;  ff*.  25.  $10,165,  1st; 
$10,326,  2d.  26.  $2300.  27.  3726.  28.  1771.  29.  $892.50. 

30.  160  rd.      31.   40.287  bbl.       32.    $74.8H.       33.  $98.       34.   $15. 
35.  2  cwt.  63  Ib.  2*f  oz. 

Page227.  — 36.  A,$12;  B,$28.  37.  N,  $140;  M,  $180.  38.  28J 
da.  39.  $400;  $500.  40.  H,  $203;  K,$290.  41.  5  da.  42.  $146.40. 
43.  $27.379;  $47.621.  44.  $1.80  per  day;  $64.80,  1st;  $73.80,  2d; 
$81,  3d.  45.  $3000. 

Page  228.  —  46.  $  186.205.  47.  24.674  A.  48.  A,  149|  bu. ; 
B,  186|  bu.;  C,  224  bu.  49.  $.7109+;  $341.232  +  .  50.  $9000. 
51.  A,  $420;  B,  $547*;  C,  $8321.  52.  833;  816.  53.  $10,  sheep; 
$30,  cow;  $90,  horse.  54.  A,  $1166f;  B,  $875;  C,  $1458i. 

55.  $840.  56.  $632.478  +  ,  A;  $666.66f,  B;  $700.854  +  ,  C. 

Page  229.  — 57.  $21.87*.  58.  7*|;  5&.  59.  60  sheep. 

60.  $.60728  +  ;  $455.46+;  $497.97  +  ;  $546.55  +  .  61.  $230.77; 
$369.23.  62.  $54331,  1st;  $50331,  2d;  $45331,  3d.  63.  $42.336; 
7056  slates.  64.  $.8179+.  65.  $2952,  D;  $738,  E;  $246,  F. 
66.  $9856,  cheaper. 

Page  230.  — 67.  $601.875;  $802.50;  $1003.125;  $5.57  per  day. 
68.  222|  panels.  69.  $288.64;  $335.87;  $419.84;  $519.55. 

70.  35ft.;  11.1408+  ft.  71.  12  men.  72.  81ft.  9.2+  in.  73.  272 
sq.ft.  74.  $400.  75.  $605,0;  $670,  D.  76.  $ 2400,  daughter ; 
$  5600,  son. 

Page  232.  —  19.  .0075.  20.  .006.  21.  .007.  22.  .0045.  23.  .00625. 
24.  .0084. 

19.  T*j.    20.  ifa.    21.  T^.    22.  jfcfo-     23.  ^    24-  ****• 

Page 234.— 2.  $146.31.  3.  $429.25.  4.  $2.882.  5.  $472.50. 
0.  $2330.  7.  $44.10.  8.  140  cows.  9.  250  A.  10.  2900  bbl. 
11.  207  sheep.  12.  $607.50. 

Page  235.  — 13.  825  A.  14.  250  boys.  15.  39.37*  T.  16.  $3966.25. 
17.  2.31yd.  18.  $168.75.  19.  $9585.  20.  $  708^  expenses ;  $1092, 
saved.  21.  $12,857.145. 

Page  236. -2.  50%.      3.  33 J%.     4.  25%.      5.  16f  %.      6.  38|%. 

7.  66§%.      8.75%.      9.  66f  %.      10.331%.      H.  30%.      12.150%. 
13.  40%.     14.  25%. 

Page  237.  — 15.  94%.  16.  14T\%.  17.74%.  18.87*%. 
19.63*%.  20.  31i%.  21.3*%.  22.  5T\  %.  23.  80f ff %. 
24.  37**%. 

Page  239.— 2.  425.     3.  2400.     4.  330.     5.  430.     6.  2394.     7.  800. 

8.  1090.     9.  1000.     10.  1416.     11.  4300.     12.  192.     13.  62.     14.  464. 
15.  625.         16.  519.        17.  3000.        18.  28*.       19.  9**.       20.  4800. 
21.11,000.        22.300.        23.  1048  bu.       24.  $26,600.       25.  $4000. 
26.  1050  bu.        27.  1475.        28.  $1200.        29.  $250.        30.  1700  T. 

31.  116  mi. 

Page  240.  —  32.  $  2610.    33.  55£  gal.     34.  $  1485.60.     35.  1984  bu. 
Page  241.  —  2.  500.      3.  600.     4.  600.      5.  700.      6.  400.     7.  600. 


ANSWERS.  453 

8.  800.     9.  1350.     10.  496.     11.  898.     12.  310  sheep.     13.  $913.043. 

14.  500  bu. 

Page  242.  — 15.  $27.30.  16.  $3801.23.  17.  $1480.  18.  $7400. 
19.  558  bu.  20.  7216.  21.  $3.63^.  22.  $6300. 

Page  243.— 2.  360.     3.  650.     4.  610.     5.  640.     6.  126.     7.  400. 

Page  244.  —  8.  $  16.48.  9.  $  1100.  10.  400  bu.  11.  $  39. 
12.  $76.25.  13.  $211.  14.  $5675.  15.  980  men.  16.  $11,250. 

17.  4760  rails.     18.  $424. 

Page  247.— 1.  $7.3395.     2.  $420.     3.  $615.     4.  $374.     5.  $301. 

6.  $315.        7.  $4248.        8.  $2906f.        9.  $100,625.       10.  $58.905. 

11.  $2382.125. 

Paee  248.  — 13.  10  %.     14.15%.     15.25%.     16.20%.     17. 

18.  50%.     19.  33 £%.    20.  25%.     21.  15%.     23.  $50.     24.  $18. 
Page  249. —25.  $1.12.         26.  $2000.        27.  $.25.        28.  $75. 

29.  $3228.     30.  $4375.     31.  $4400.     32.  $18,250.     35.  $35.60. 

Page  250.—  36.  $1125.  37.  $212.669  +  .  38.  $113.70.  39.  $380. 
40.  $1089.20.  41.  $6790.50.  42.  $4740.932  +  .  43.  14f%. 
44.  $83.375.  45.  $  1  per  yd.  46.  $480  loss.  47.12%. 

Page  252.— 3.  $44.10.      4.  $18.70.       5.  $3277.50.       6.  $7500. 

7.  8552  bu.  1  pk.  7.3  qt.     8.  $2698.31|-. 

Page  253.— 9.    $4506.341  +  .        10.    16,000  bu.        11.    $1793.60. 

12.  $3450.     13.  3£%. 

Page  254.— 1.  $324.  2.  $516.80.  3.  $598.50.  4.  $570.96. 
5.  $7.08|.  6.  $3.876.  7.  $208.25.  8.  $273.52.  9.  $203.853+. 
10.  $397.80.  11.  $9.60.  12.  $38.304.  13.  $301.53.  14.  $283.93£. 

15.  $12.793+.     16.     $269.325.     17.  $289.80;  19£%. 

Page  257.  —  3.  $47.4538.  4.  $105.8565.  5.  $58.49.  6.  $110.15685. 
7.  .0055.  8.  $185.835.  9.  .0065.  10.  $20.475,  A;  $27.30,  B  ; 
$36.3675,  C.  11.  $6420. 

Page  258.  — 1.  $183.75.  2.  $59.85.  3.  $1161.207.  4.  $12.41. 

5.  $207.48.  6.  $1890.  7.  $16.65.  8.  $1848.  9.  $28.50. 
10.  $580.50.  11.  $852. 

Page  260.  — 1.  $233.75.  2.  $54.  3.  $525.  4.  $710.50. 

5.  1H%.  6.  $1271.605.  7.  $57.40.  8.  $30,000.  9.  $5.225. 

Page  261.  — 1.  $102.80.       2.  $191.75.  3.  $2011.95.      4.  $35. 

5.  $3739.20.     6.  $4208.     7.  $34.30. 

Page  263.—  2.  $198.736.    3.  $109.06.    4.  $162.528.    5.  $423.572. 

6.  $60.12.     7.  $210.364.     8.  $161.245.     9.  $417.294.     11.  $144.343. 
12.  $133.3864.     13.  $117.607.        14.  $118.813.          15.    $210.0564. 

16.  $208.227. 

Page  264.  — 17.  $380.298.  18.  $150.191.  19.  $22.527.  20. 
$31.425.  21.  $45.143.  22.  $94.43.  23.  $119.028.  24.  $154.677. 
25.  $268.315.  26.  $541.477.  28.  $42.32;  $355.82.  29.  $223.80; 
$1159.55. 

Page  265.— 30.  $35.18;  $304.68.  31.  $47.85;  $516.60. 

32.  $42.71;  $316.79.  33-  $49.61;  $414.11.  34.  $78.83;  $364.92. 
35.  $95.72;  $464.47.  36.  $100.14;  $468.32.  37.  $265.66; 

$  846.56.  38.  $  24.69  ;  $  300.29.  39.  $  73.44  ;  $  541.69.  40.  $  303.69  ; 
$1118.96.  41.  $14.85;  $140.65.  42.  $17.96  ;  $202.46.  43.  $135.08; 
$695.33.  44.  $62.93;  $439.40.  45.  $254.92;  $1254.92. 


454 


ANSWERS. 


5.  $22.52.      6.  $22.195. 
4.  $43.14.      5.  $297.69. 


46.  $1091.80;  $5211.80.        47.  $366.05 ;  $3546.05.       48.  $856.27  j 
$3731.27,     50.  $462.03.     51.  $356.87.     52.  $442.19.      53.  $703.16 
54.  $736.61. 
Page  266.  — 2.  $75.38.        3.  $88.24.        4.  $58.92.       5.  $65.51. 

6.  $121.11.      7.  $72.95.       8.  $191.94.      9.  $132.89. 

Page  267.  — 10.  $62.89.     11.  $142.14.     12.  $182.98.     13.  $417.03. 

14.  $221.39. 

2.  $9.10.        3.  $11.24.        4.  $19.63. 

7.  $19.60.     8.  $12.51.     9.  $109.50. 
Page    268.— 2.  $22.89.       3.  $48.36. 

6.  $54.04.       7.  $38.46.       8.  $66.30.       9.  $175.39. 

1.  $6.52.    2.  $7.49.     3.  $10.47.    4.  $12.41.   5.  $59.61.  6.  $260.69. 

Page  269.  — 2.  $1457.28.  3.  $1844.30.  4.  $2668.15.  5.  $2427.63. 
6.  $3002.63.  7.  $3565.00.  8.  $5208.56.  9.  $5015.325.  10.  $7173.07. 
11.  $13088.54.  12.  $10189.18. 

Page  270.  — 2.  $37.08.      3.  $83.055.      4.  $53.32. 

Page  271.  — 5.  $113.205.  6.  $139.51.  7.  $140.72.  8.  $41.41. 
9.  $84.28.  11.  $126.39.  12.  $76.21.  13.  $230.61.  14.  $351.50. 

15.  $96.69.     16.373.525.     17.  $520.855.     18.  $645.19.    19.  $204.80. 
20.  $125.05.  21.  $1467.43.  22.  $1417.96.          23.  $21.74. 
24.  $362.65.       25.  $236.40.       26.  $254.68.      27.  $1898.369. 

Page  275.  — 8.  $18.725.     9.  $27.23.      10.  $43.38.      11.  $52.08. 
Page  276.  — 1.  $251.54.    2.  $300.20.     3.  $1665.31.     4.  $433.54. 
Page  278.  — 2.  $366.02.     3.  $495.83.     4.  $452.41.     5.  $261.15. 
6.  $520.81.     7.  $581.85.     8.  $1210.26.     9.  $1307.14. 
Page  279.  — 2.  4%.     3.6%.     4.7%.    5.6%.      6.  2|£%.     7.6%. 

8.  6J%.     9.  6%.     10.  5%. 

Page  280.  —  2.  2  yr.  6  mo.  3.  5  yr.  1  mo.  27  da.  4.  2  yr.  2  mo. 
18  da.  5.  7  yr.  8  mo.  6.  5  yr.  9  mo.  7.  3  yr.  8  mo.  8.  3  yr.  8  mo. 
16  da.  9.  14f  yr.  10.  12  yr.  6  mo.  11.  3  yr.  6  mo.  12.  18T2T  yr. 
13.  Oct.  1,  1891.  14.  June  24,  1875. 

Page  281.  — 2.  $255.  3.  $250.  4.  $219.30.  5.  $1292.21. 
6.  $1279.32.  7.  $538.95.  8.  $3392.59.  9.  $2098.72.  10.  $297.19. 
11.  $2273.39.  12.  $1693.33.  13.  $13,769.23. 

Page   283.— 2.    $551.91;     $24.84.  3.    $730.42;     $30.43. 

4.  $395.93;  $41.57.  5.  $584.32;  $64.28.  6.  $1108.03,  $91.97; 
$1122.37,  $77.63.  7.  $1455.93,  $152.07,  $1491.19,  $116.81. 

8.  $2488.42,  $86.58;  $2499.63,  $75.37.  9.  $1331.23,  $26.62; 
$1332.86,  $24.99.  10.  $2822.20,  $358.30.  11.  $148.65.  12.  $9.71. 
13.  $440.47.  14.  $4.13. 

Time. 

Page  286.  —  3.  May      4 

4.  Aug.     14 

5.  Dec.       5 

6.  May      8 

7.  Sept.     2 
Page  287.—  8.  Oct.   24/27 

9.  Oct.      5/s 

10.  July     Vs 

11.  June  16/i9 
1£.  Nov.     V* 


Term  of  dis. 

Discount. 

Proceeds. 

55  days 

$7.01 

$757.74. 

81  days 

7.26 

530.19. 

34  days 

4.82 

845.68. 

26  days 

1.02 

234.66. 

96  days 

8.82 

463.66. 

47  days 

6.53 

993.47. 

64  days 

11.92 

1105.13. 

46  days 

1.05 

135.87. 

76  days 

9.65 

643.46. 

63  days 

16.80 

1183.20. 

ANSWERS.  455 

Page  288.— 2.  $600.       3.  $340.       4.  $1882.68.      5.  $1010.61. 

6.  $909.14. 

Page  292. —2.  $13,368.75.     3.  $7761.50.    4.  $2480.625. 

Page  293.— 5.  $4065.     6.  $6502.50.     7.  $14,203.125.     8.  $1305. 

9.  $7766.875.        10.  $42,328.125.        11.  $14,562.50.        12.  $422.50. 
14.  120  shares.       15.  112  shares.       16.  160  shares. 

Page  294.  — 18.  $300.  19.  $280.  20.  $1090.  21.  The 
latter  $20.  22.  $720. 

Page  295.— 24.  $12,195.  25.  $22,740.  26.  $27,309.375. 

27.  $19,125.  29.8%.  30.  6|%.  32.  142f%.  33.150%. 

Page  298.  — 1.  48,600.        2.  $4896.        3.  $5600.       4.  $17,500. 

5.  61$%.       6.  $3041.25.       7.  28^%-       8.  12*%. 

Page  299.  —  9.  10  %.  10.  $  12,000 ;  $  180.  11.  $  1302.40.  12.  $  7855. 
13.  2£%.  14.  $15,000.  15.  $12,580.15.  16.  .004 rate.  17.  $951. 
IB.  $292.50.  19.  $185.79.  20.  $21,016.67. 

Page  300.—  21.  $378.48.  22.  $180.78.  23.  $281.25.  24.  $10.96. 
25.  $167.05.  26.  $273.85.  27.  $  8.331  28.  $5715.  29.  $360. 
30.  18f$%.  31.  Increased  $  120. 

Page  304.  — 3.  $1209.      4.  $1494.375.     5.  $805.      6.  $1581.60. 

7.  $472.98. 

Page  305.  — 8.  $588.45.    9.  $890.10.    10.  $1165.20.    11.  $551.42. 

12.  $1696.48.     15.  $2779.16.     16.  $1244.44. 

Page  306.  — 17.  $1811.32.  18.  $1991.04.  19.  $648.70. 

20.  $1027.57. 
Page  307.  — 3.   $2073.83.        4.   $1557.09.        5.   £400  13s.  8  +  d. 

6.  £589185.  l  +  d.      7.   £  822  3s.  11  +  d.      8.   $968.99.     9.   $1518.34. 

10.  $1269.20. 

Page  309.— 2.  A,  $1600;  B,  $900;  C,  $700.         3.  A,  $765.96; 

B,  $919.15  ;  C,  $714.89.    4.  A,  $3490.91 ;  B,  $5585.45  ;  C,  $3723.64. 
Page  310.  —  5.  A,  $  1631.25  ;  B,  $  1522.50  ;  C,  $2066.25  ;  D,  $  1740. 

6.  A,  $1833^ ;  B>  $2166|.      7.  A,  $7507.50;  B,  $8872.50.       8.  E, 
$  1080  ;  F,  $  1260  ;  G,  $  1440 ;  H,  $  1620.       9.  A,  $  1250  ;  B,  $  1875  ; 

C,  $3125. 

Page  311.  — 11.  A,  $840;  B,  $1152;  C,  $960.  12.  A,  $1040; 
B,  $540.  13.  A,  $19.20;  B,  $32;  C,  $76.80.  14.  A,  $10,270;  B, 
$  10,270  ;  C,  $  5135.  15.  D,  $  218.18  ;  E,  $  174.55  ;  F,  $ 87.27.  16.  A, 
$990;  B,  $1485;  C,  $1155.  17.  A,  $1242.69;  B,  $1257.31. 

Page  318.  — 3.  $200.       4.  $407.22.       5.  12f  da.       6.  67,200ft. 

7.  135ft.    8.  $3.15.    9.  $175.    10.  16|  bu.    11.  65  bbl.     12.  520  bu. 
Page  319.— 13.  $56.59.      14.  31$  yd.      15.  10  men.      16.  58f  ft. 

17.  40  revolutions.  18.  22$  da.  19.  46$  da.  20.  125.  21.  20  da. 
22.  20  h.  41.44  min.  23.  2711}  bu.  24.  214T7^  mi. 

Page  321.  — 2.  9  men.  3.  18  days.  4.  216  men.  5.  $880 
6.  19,800  Ib.  7.  289f  bu.  8.  $685.71. 

Page  322.  — 9.  214  da.      10.  8A  da.       11.  750  Ib.       12.  21^  da. 

13.  158|4  da.     14.  410A-  yd.     15.  27,600  Ib.     16.  18,750  Ib.    17.  21$-  T. 
Page  323.  —  3.  60  ;  150  ;  210.    4.  50  ;  100  ;  150  ;  200  ;  250.    5.  216  ; 

144  ;  108.    6.  $  480  ;  $  540  ;  $  576.    7.  $  8400  ;  $  8750  ;  $  9000.    8.  Wife, 
$60,344.82  ;  each  son,  $43,103.45;  each  daughter,  $34,482.76. 
Page  325.  —2. 169;  324;  441 ;  1296.   3.  729;  3375;  13,824;  74,088. 


456  ANSWERS. 

4.  625  ;  1024  ;  2304  ;  4356.    5. 10,648  ;  15,625  ;  157,464  ;  357,911.   6.  256 ; 
1296  ;  20,736  ;  130,321.    7.  12,167  ;  27,000  ;  79,507  ;  421,875.   8.  2025  ; 
4761  ;  7396  ;  8836.     9.  $  ;  #  ;  ff  ;  f f .     10.  *  ;  ^  ;  f $  ;  T\V     11.  7225. 
12.  117,649.     13.  1600.     14.  125,000.     15.  10,404.     16. 42.25.     17.  .5625. 
18.  .000125.     19.  .000016.     20.  9.3025.     21.  f||.    22.  T6A\.     23. 

24.  Hf.     25.  ffif.     26.  56 J.     27.  413f.     28.  16.40£.  "  29.  12.2 
30.  25.2255TV 
Page  327.  — 2.  15;  36;  49.        3.  105;  120;  216.      4.  7;  12;  26. 

5.  27  ;  32  ;  42.     6.  16  ;  12.     7.  24  ;  12. 

Page  331.— 4.  55.      5.  74.      6.  98.      7.  109.      8.  115.       9.  121. 

10.  145.     11.  153.     12.  204.     13.  229.     14.  278.     15.  416.    16.  43.6. 
17.  .717.     18.  9.89.     19.  .035.     20.  9.469.     21.  50.405.     22.  531,441. 
23.  ff .    24.  f£f .    25.  |i|.     26.  49.5.   27.  2214.    28.  ffi.    29.  4.24264. 
30.  5.91607.     31.  6.85565.     32.  9.53939.     33.  11.26942.     34.  15.65247. 
35.  19.23538.     36.  28.51315.      37.  2.69258.      38.  .72525.     39.  .17606. 

40.  6.54980.     41.  3.56195.     42.     5.00749.    43.  .06831.    44.  .02108. 
Page  332.  —  1.  85  rd.     2.  80  rd. ;  160  rd.     3.  88  men.    4.  972  yd. 

5.  $40.50. 

Page  333.— 7.  41.56+  ft.  8.  123.59+  ft.  9.  $210.  10.  339.41 
rd.  11.  56ft.  12.  32.31+ ft.  13.  142.13ft. 

Page  334.— 2.  16  ft.  3.  53.66  rd.;  40.24  rd.  4.  18.178  rd. 
5.  480  gal.  6.  9:36. 

Page  340.  —  5.  38.      6.  56.      7.  74.      8.  95.      9.  145.       10.  352. 

11.  507.       12.  1231.       13.  1596.       14.  4567.       15.  7894.       16.  23.7. 
17.  3.04.     18.  .055.     19.  .0125.    20.  1.2599.     21.  1.81712.     22.  .7631 ; 

*7Fa3!e'341.  —  1.  38  in.  2.14ft.  3.  3249  sq.  in.  4.4ft.  5.11ft. 
6.16  in.  6.  lift.  10.77  in.  7.  $27.52.  8.13.55ft.  9.  Rectangle, 
169.616  sq.  ft.  greater. 

Page  342.  —  2.  64.  3.  32Ulb.  4.  803.52  bu.  5.2.38ft.  6.  3.76  in. 
7.  I81|oo9lb>  8  5.73  in.  9>  26ft.  10.  Depth,  5.039  in.;  diam.,1 1.654  in. 

Page  350.  — 1.  15,378.  2.202,700.  3.  .00230728.  4.  .00413+. 
5.  1£.  6.  $25.20.  7.  $143.38.  8.  A,  $54;  B,  $36.  9.  15TV  yd. 
10.  $11.635.  11.  $4.38.  12,  $500.  13.  Girl,  $1.00;  boy,  $.75. 

Page  351.  — 14.  8  hr.  48  min.  15.  4f  f  da.  16.  $14.78.  17.  150 
oranges.  18.  $22,200.  19.  Wheat,  120  bu. ;  oats,  160  bu.;  corn,  200 
bu.  20.  694  A.  74.78  sq.  rd.  21.  121  cords.  22.  $  28,400.  23.  26 U 
ft.  24.  50f  oz.  25.  1584.  26.  690. 

Page  352.— 27.  $12,480.  28.  $24,527.16.  29.  44  ft.  per  sec. 
30.  A,  $55;  B,  $48  ;  C,  $52.  31.  $80.05.  32.  $1500.  33.  134  oz. 
34.  51^- da.  35.  $4.70.  36.  $147.06. 

Page  353. —37.  $40. 33|  loss.    38.165ft.    39.  $.314.    40.  $180.29. 

41.  810  rev.    42.  252  A.    43.  $1057.268.    44.  24,897.6  mi.    45.  $2500. 
Page  354.— 46.  1734  sq.  in.     47.  19  horses.     48.  Wheat,  $1.50; 

corn,  $.40.  49.  $31.35.  50.  $21.60.  51.  $152.36.  52.  $46.20. 
53.  $882.46.  54.  3750  sq.  ft.  55.  106|  acres.  56.  16|  yr.  57.  30  ft.  ' 
Page  355.— 58.  $152.  59.  106|.  60.  72.32  bu.  61.  Nothing. 
62.  All,  18fff  da.  ;  A,  58ff  da.  ;  B,  70£f  da.  ;  C,  46^  da.  63.  19|| 
bbl.  64.  $776.53|.  65.  120  da.  66.  30  min.  67.  $400  lost.  68.  81 
shares.  69.  $73.75. 


ANSWERS.  457 

Page  356.  —  70.  23$  mi.  71.  $177.10.  72.  24  min.  73.  10° 
Centigrade.  74.  $9975.  75.  $1203.  76.  $454.20.  77.25.09ft. 
78.  A,  $437  ;  B,  $414. 

Page  357. —79.  12  oz.  80.  20.003$.  81.  $7210.  82.  467.5736 
acres.  83.  $.557.  84.  32  oz.  85.  299.025  A.  »6.  27  Galls.  87.  53 
da.  88.  $2964.71. 

Page  358.— 89.  1st,  $11,200  ;  2d,  $11,025  •  3d,  $10,300.  90.  $208$. 
91.  $01.60.  92.  $1775.  93.  $1148.35.  941  $935.55.  95.  1st,  68f 
da.  ;  2d,  53$  da.  96.  $2875.  97.  A,  $432  ;  B,  $405  ;  C,  $459. 

Page  359. —98.  $312.  99.  12G.o'8  ft.  100.  $390.  101.  46  min. ; 
5  times.  102.  Daughter,  $813.38;  son,  $686.62.  103.  A,  $450; 
C,  $750.  104.  Lot,  $1650;  store,  $2200;  house,  $7150. 

Page  360.  — 105.  $  343b.  2d5.  106.  150  Ib.  107.  25  rd.  108.  40  rd. 
109.  Horse,  $240;  carriage,  $ 270.  110.  $66|.  111.  10%.  112.  $3.66f. 
113.  3  mo.  114.  45.48ft. 

Page  361.  — 116.  1st,  162;  2d,  144;  3d,  128.  116.  9.801  da. 

117.  24ft.  lib.  ^320. 185.  119.  3  yr.  5  mo.  7  da.  120.  6%. 
121.  A,  $  1344  ;  B,  $  1008  ;  C,  $  840.  122.  D,  $  126  ;  E,  $  120  ;  F,  $  135. 
123.  A,  $7500  B,  $6000;  C,  $5100.  124.  15f%. 

Page  362.  — 125.  20  ft.  126.  15.6  oz.  127.  62.  128.  57,330. 
129.  A,  $'i425;  B,  $1900;  C,  $1425.  130.  $2600.  131.  Lost,  $40; 
6£%.  132.  4  hr.  56  min.  12  sec.  133.  5  hr.  5  min.  32  sec. 

134.  Gold,  42$  gr.  135.  .006J.  136.  $500. 

Page  363.  — 137.  27.64ft.  138.  $233.99.  139.  $9.228. 

140.  $64,931.  141.  13.775  A.  142.  John,  81  da.  ;  Charles,  101£  da. 
143.  A,$3;B,$21.  144.  A,50da.;  B,60da.  145.  $74.07.  146.  $5000. 

Page  364.— 147.  $361.07.  148.  .131  in.  149.  50  min.  150.  37$%. 
151.  8oz.l8pwt.l4$gr.  152.  $550.  153.  $46,710.  154.  $22.22.  155.  $14. 

Page  365.  — 156.  $700.  157.  21.  158.  $3206.63.  159.  Wheat, 
$2000  ;  barley,  $3000  ;  oats,  $4000.  160.  $557.244.  161.  Lost,  $10. 
162.  1501b.  ;  90  Ib.  163.  $600. 

Page  366.— 164.  Better,  $4650;  other,  $3100.  165.  $25.40. 
166.  Former,  $%.  167.  $25,000.  168.  54ff%.  169.  266,951.87yd. 
170.  A,  $1200;  B,  $1600;  C,  $800.  171.  A,  $7200;  B,  $6000;  C, 
$  5040. 

Page  367.  — 172.  200  shares.  173.  Former,  T2^%.  174.  Yes,6|%. 
175.  $3827.96.  176.  Horse,  $391.72  ;  carriage,  $195.86.  177.  $22.26. 
178.  4448  slates  ;  $211.28.  179.  $15.  180.  1st,  3.68  in.  ;  2d,  4.78  in.  ; 
3d,  11.54  in. 

Page  369.  —  2.  2  mo.  29  da.       3.  2  mo.  18  da.       4.  1  mo.  12  da. 

5.  2  mo.  10  da.     6.  3£  mo. 
Page  370.  — 2.  June  20,  1887. 

Page  371. —3.    May  1,  1892.     4.  Jan.  24,  1892.     5.   Dec.  22,  1891. 

6.  April  23,  1892.    7.  June  7,  1892.    8.  Aug.  16,  1892.    9.  5$  months. 
Page  373.— 2.  June  20,  1892.    3.  June  15,  1892.    4.  July  6,  1892. 

5.  $1198.80. 

Page  375.  —  2.    $329.90.     3.    $177.94. 

Page  377.— 2.    $1.72.     3.    209T^  ft.     4.    5070. 

Page  378.  —  5.  55.  6.  198.  8.  3775.  9.  505.  10.  330  mi. 
11.  78.  12.  $3360.  13.  1287$. 


458  ANSWERS. 

Page  379.  —  2.  2430.   3.10,240.  4.  $984.15.  5.  $133.82.   6.  $696.85. 

Page  380.—  8.  1364.    9.  42V      10.4.    11.  *£&$£•    12.  $20,971.51. 

Page  381.  —  2.  $  5304.387.     3.  $  3638.85.     4.  1 2046.58  ;  $  3022.35. 

Page  382.  —  2. 6%.     3.4%.    4.7%.    5.8%. 

2.  15  yr.  8  mo.  18  da.  3.  14  yr.  2  mo.  12  da.  4.  10  yr.  10  mo. 
3  da.  5.  11  yr.  2  mo.  18  da.  6.  10  yr.  4  mo.,  nearly. 

Page  384.— 3.  $3488.  4.  $8486.40.  5.  $9810.81.  6.  $30,780. 
7.  $1173.91. 

Page  385.—  2.  $1609.49.  3.  $3447.39.  4.  $9710.35.  5.  $17,465.74. 
6.  $1839.28.  7.  $4060.55.  8.  $4024.24.  9.  $5827.34. 

Page  388.  —  2.  28.  3.17.  4.72.  5.63.  6.28.  7.544.  8.73. 
9.  15.  10.  42.  11.  f.  12.  ||.  13.  |f.  14.  f.  15.  4.  16.  TV 
17.  TV  18.  tf|.  19.  fi.  20.  *.  21.  ff.  22.  fff.  23.  48. 

Page  391.— 17.  1260.  18.3584.  19.6160.  20.33,228.  21.5616. 
22.  156,240.  23.  7770.  24.  34,650.  25.  27,720.  26.  60  qt.  27.  270 
in.  28.  180  min.  A,  15 ;  B,  12  ;  C,  10.  29.60yd. 

Page  392.  — 1.  &.  2.  Tfff.  3.  Tf?.  4.  A-  5.  ft.  6.  ^  7.  413  rails. 

Page  393.  —  1.  i£&.  2.  \*£.  3.231.  4.-6/.  5.^^.  6.100.  7.95isec. 
.  15. 


Page  396.- 14.  ^.  15.  ^V  16.  ^.  17.  ff£§.  18. 
19.  fVVo-  20.  TVvV  21.  ffi£.  22.  iff.  23.  ^3o-  24.  T§- 
25.  fSSfft- 

Page  399.— 2.  110,244;  141,421;  110,333.  3.  21,024;  25,643; 
15,316.  4.  331,223;  1,110,333;  330,032.  5.  3701;  4*63;  403*. 

2.  4354  ;  731 ;  4858 ;  12,884.  3.  492  ;  33,511 ;  2714  ;  5534.  4.  3549  ; 
76,295  ;  8831 ;  53,123. 

1.  30,3335.  2.  30,314..  3.  27,5649.  4.  24,564..  5.  31,650a. 
6.  2,034,4315. 

Page  405.  —  1.  31.416  sq.  ft.  2.  40  sq.  ft.  3.  144  sq.  ft. 
4.  37.6992  sq.  ft. 

Page  406.  —  1.  540  sq.  ft.  2.  376.992  sq.  ft.  3.  628.32  sq.  ft. 
4.  $64.  5.  89.5356  sq.  ft.  6.  400  sq.  ft.  7.  75.3984  sq.  ft 
8.  157.08  sq.ft. 

Page  407.  — 1.  251.328  sq.  ft.  2.  3000  sq.  ft.  3.  $5.654. 
4.  340  sq.  ft.  5.  $33.68. 

1.  4.908  sq.  ft.    2.  1.396  sq.  ft.     3.  26.50+  sq.  in.     4.  45.836  sq.  ft. 

Page  408.  — 1.   2  cu.  ft.  2.   7.0686  cu.  ft.  3.   $9.00. 

4.  462.852+  bu.      5.  2632.089+  gal.       6.   $4013.80+. 

Page  409.  —  1.  84.8232  cu.  ft.     2.  18,000  cu.  ft.     3.  12,440.736  Ib. 

4.  7296  Ib.     1.  4666§  cu.  ft.       2.  236.405  +  cu.  ft. 
Page  410.  — 3.  136.136  cu.  ft.     4.  6415.718  gal. 

1.  65.45  cu.  ft.     2.  268.0832  cu.  ft.     3.  14.1372  cu.  ft.     4.  795.217+  Ib. 

5.  8.181  cu.  ft.    6.  8181.25  cu.  ft.    7.  29.4674+  Ib. 

Page  414.  — 1.  .15675  Km.  2.  75.60  a.  3.  67.341.;  .6734  HL 
4.  43.628  g.;  .043628  Kg.  5.  7501.;  7.5  HI.  6.  .087637  sq.  m.; 
.00087637  a.  7.  230.5  g.  8.  4500  Dg.  9.  1300  Kg.  10.  1000  Kg.; 
1000  1.  11.  65.75075  cu.  m.;  65,750.75  Kg.  12.  810  Kg. 

Page  415.  — 13.  7210  g.  14.  920  Kg.  15.  Former  17  ^  per  m. 
16.  27.032  a.  17.  22.26  sq.m.  18.31201.  19,  .90225m.  20.88.9042 
Kg.  21.  225  HI.  22.  337.5  g.  23.  10,9441.  24.  30.8965  hr. 
25.  292.948  Kg.  26.  $32.266.  27.  6.592  cu.  yd.  28.  $29.918. 


ANSWERS.  459 

Page  416.  —  29.  $243.036.  30.  2.583  Kg.  31.  $27.215+. 
32.  92.986+  Hg.  33.  364.544+  revolutions.  34.  21  1)1.  is 

12.476  gal.  greater.  35.  2.204+  Ib.  Avoir.,  2.670+  Ib.  Troy. 

36.  3785.37  g.  37.  $.997  per  Ib.  38.  .0929  sq.  m.  39.  9.29+  ca. 
40.  95.9+  Ib.  41.  28.327  Kg.  42.  3423.451+  m.  43.  63.4+  gal. 
44.  $.049+ per  Ib.  45.  4755. 15  gal.  46.  $2.326- per  bu.  47.  88  Hg. 

Page  417.  —48.  580.598+  Kg.  49.  2.5  sp.  g.  50.  30184  Dg. 
51.  $.171+ per  qt.  52.  26.729+ g.  53.  .245mm.  54.  $1.92. 
55.  13,500,000  g.  56.  118.371  +  g.  57.  .83£  m.  58.  $1.261872 
per  yd.  59.  1.0416  +  Ib.  60.  30.283  -  francs  per  gal.  61.  $1680. 
62.  2.2046  Ib. 

Page  433.  — 2.  $1685.66. 

Page  434.  —  3.  $952.64.  4.  $696.61.  5.  $399.59.  6.  $687.79 
7.  $842.87.  8.  $2114.64. 

Page  435.  —  2.  $  1881.38.      3.  $  3884.60. 

Page  437.  — 2.  $266.01.         3.  $242.799. 

Page  438.  —  3.  $  63.14.  4.  $  4.05.  6.  $22,471.63  grand  list,  $  .90. 
6.  $92.25. 


NEW   SERIES  OF  THE 
NATURAL     GEOGRAPHIES 

REDWAY  AND    HINMAN 

TWO  BOOK  OR  FOUR  BOOK  EDITION 

Introductory  Geography     .   $0.60          School  Geography  .     .      .   $1.25 
In  two  parts,  each     .      .      .40  In  two  parts,  each     .  .75 


IN  the  new  series  of  these  sterling  geographies  emphasis  is  laid 
on  industrial,  commercial,  and  political  geography,  with  just 
enough  physiography  to  bring  out  the  causal  relations. 
^[  The  text  is  clear,  simple,  interesting,  and  explicit.  The 
pictures  are  distinguished  for  their  aptness  and  perfect  illus- 
trative character.  Two  sets  of  maps  are  provided,  one  for 
reference,  and  the  other  for  study,  the  latter  having  corre- 
sponding maps  drawn  to  the  same  scale. 
«ff  The  INTRODUCTORY  GEOGRAPHY  develops  the 
subject  in  accordance  with  the  child's  comprehension,  each 
lesson  paving  the  way  for  the  next.  In  the  treatment  of  the 
United  States  the  physiographic,  historical,  political,  industrial, 
and  commercial  conditions  are  taken  up  in  their  respective 
order,  the  chief  industries  and  the  loc  lities  devoted  largely  to 
each  receiving  more  than  usual  consideration.  The  country 
is  regarded  as  being  divided  into  five  industrial  sections. 
^j  In  the  SCHOOL  GEOGRAPHY  a  special  feature  is 
the  presentation  of  the  basal  principles  of  physical  and  general 
geography  in  simple,  untechnical  language,  arranged  in  num- 
bered paragraphs.  In  subsequent  pages  constant  reference  is 
made  to  these  principles,  but  in  each  case  accompanied  by 
the  paragraph  number.  This  greatly  simplifies  the  work, 
and  makes  it  possible  to  take  up  the  formal  study  of  these 
introductory  lessons  after  the  remainder  of  the  book  has  been 
completed.  With  a  view  to  enriching  the  course,  numerous 
specific  references  are  given  to  selected  geographical  reading. 


AMERICAN    BOOK     COMPANY 

c«o 


APPLIED    PHYSIOLOGIES 

By    FRANK    OVERTON,    A.M..     M.D.,    late    House 
Surgeon  to  the  City  Hospital,   New  York  City 


Primary  Physiology  .     .     .   $0.30         Intermediate  Physiology     .  $0.50 
Advanced  Physiology    .     .  $0.80 


OVERTON'S    APPLIED    PHYSIOLOGIES   form  a 
series  of  text-books   for  primary,  grammar,  and  high 
schools,  which    departs    radically   from    the    old-time 
methods  pursued  in  the  teaching  of  physiology.     These  books 
combine  the  latest  results  of  study  and  research  in  biological, 
medical,  and   chemical   science  with    the   best  methods  of 
teaching. 

^y  The  fundamental  principle  throughout  this  series  is  the 
study  of  the  cells  where  the  essential  functions  of  the  body 
are  carried  on.  Consequently,  the  study  of  anatomy  and 
physiology  is  here  made  the  study  of  the  cells  from  the  most 
elementary  structure  in  organic  life  to  their  highest  and  most 
complex  form  in  the  human  body. 

^[  This  treatment  of  the  cell  principle,  and  its  development 
in  its  relation  to  life,  the  employment  of  laboratory  methods, 
the  numerous  original  and  effective  illustrations,  the  clearness  ol 
the  author's  style,  the  wealth  of  new  physiological  facts,  and  the 
logical  arrangement  and  gradation  of  the  subject-matter,  give 
these  books  a  strength  and  individuality  peculiarly  their  own. 
^[  The  effects  of  alcohol  and  other  stimulants  and  narcotics 
are  treated  in  each  book  sensibly,  and  with  sufficient  fullness. 
But  while  this  important  form  of  intemperance  is  singled  out, 
it  is  borne  in  mind  that  the  breaking  of  any  of  nature's  laws 
is  also  a  form  of  intemperance,  and  that  the  whole  study  of 
applied  physiology  is  to  encourage  a  more  healthful  and  a 
more  self-denying  mode  of  life. 

^[  In  the  preparation  of  this  series  the  needs  of  the  various 
school  grades  have  been  fully  considered.  Each  book  is  well 
suited  to  the  pupils  for  whom  it  is  designed. 


AMERICAN    BOOK     COMPANY 


COMPOSITION-RHETORIC 

(Steps  in  English  Series) 

By  THOMAS  C.  BLAISDELL,  Ph.D.,  Professor  of  English, 
Michigan  State  Agricultural  College 

$1.00 


THIS  book,  which  aims  to  teach  young  people  to 
write  effectively,  is  suited  for  use  in  any  secondary 
school.  Its  ingenious  method  of  treatment,  its  fresh 
and  interesting  character,  its  great  simplicity  and  suggestive- 
ness,  will  prove  stimulating  and  inspiring  to  every  student. 
The  work  lays  a  foundation  for  the  appreciation  of  literature. 
^j  Models  from  the  master  writers  are  furnished  and  pupils 
are  asked  to  use  their  own  experiences  as  working  material. 
They  are  taught  to  write  accurately  by  being  trained  to 
recognize,  and  thus  to  avoid,  their  errors.  Principles  are 
studied  only  when  they  are  encountered,  each  pupil  being 
obliged  to  learn  merely  those  of  which  he  is  ignorant. 
*U  The  most  important  qualities  which  characterize  literature 
are  each  taken  up  in  turn  and  considered.  Selections  from 
the  works  of  famous  writers  are  inserted  at  frequent  intervals 
for  purposes  of  illustration,  and  it  is  shown  by  analysis  how 
they  appeal  to  the  feelings,  and  why  they  attain  the  various 
results  necessary  to  an  interesting  expression  of  their  thoughts. 
The  student  is  taught  that  literature  is  full  of  suggestion,  and 
he  is  made  to  understand  the  various  devices  by  which  an 
author  conveys  this  suggestion., 

5[  When  ihese  methods  have  been  discovered  and  sufficiently 
illustrated,  the  learner  is  asked  to  use  them  in  writing  about 
familiar  experiences.  In  these  exercises,  which  are  very 
numerous,  while  accuracy  of  expression  is  sought,  fluency  of 
expression  is  considered  of  chief  importance.  At  first  com- 
positions of  only  a  few  paragraphs  in  length  are  required,  but 
later  the  character  sketch,  the  short  story,  and  the  essay  are 
taken  up.  Letter  writing  is  emphasized  throughout  the  book. 


AMERICAN     BOOK    COMPANY 

(85) 


THE   GATEWAY  SERIES 

of  English  Texts  for  Admission  to  College 

Henry  Van  Dyke,  General  Editor 


ADDISON'S  Sir  Roger  de  Coverley  Papers  (Winchester)  $0.40 
BURKE' s  Speech  on  Conciliation  (MacDonald)      .  .35 
BYRON,  WORDSWORTH,  SHELLEY,  KEATS,  AND  BROWN- 
ING— Selections  (Copeland  and  Rideout)  ...  .40 

CARLYLE'S  Essay  on  Burns  (Mims) .35 

COLERIDGE'S  Rhyme  of  the  Ancient  Mariner  (Wood- 
berry)      30 

EMERSON'S  Essays — Selections  (Van  Dyke) 35 

FRANKLIN'S  Autobiography  (Smyth)  .....  .40 

GASKELL'S  Cranford  (Rhodes) .40 

GEORGE  ELIOT'S  Silas  Marner  (Cross) .40 

GOLDSMITH'S  Vicar  of  Wakefield,  and  the  Deserted 

Village  (Tufts) .45 

IRVING'S  Sketch-Book — Selections  (Sampson)    ...  .45 

LAMB'S  Essays  of  Elia — Selections  (Genung)  ....  .40 

MACAULAY'S  Addison  (McClumpha)  .....  .35 

MACAULAY'S  Milton  (Gulick) -35 

MACAULAY'S  Addison  and  Johnson  (McClumpha  and 

Clark) .45 

MACAULAY'S  Life  of  Johnson  (Clark) .35 

MILTON'S  Minor  Poems  (Jordan) .35 

SCOTT'S  Ivanhoe  (Stoddard)      ....  .50 

SCOTT'S  Lady  of  the  Lake  (Alden) .40 

SHAKESPEARE'S  As  You  Like  It  (Demmon)  ....  .35 

SHAKESPEARE'S  Julius  Caesar  (Mabie) .35 

SHAKESPEARE'S  Macbeth  (Parrott) .40 

SHAKESPEARE'S   Merchant  of  Venice    (Schelling)   .  .35 

TENNYSON'S  Idylls  of  the  King — Selections  (VanDyke)  .  3  5 

TENNYSON'S  Princess  (Bates)    ....               ,  .40 


Teachers'  Outlines  for  Studies  in  English,  with*  refer- 
ences to  the  Gateway  Series  (Blakely)  ....         .50 

AMERICAN    BOOK    COMPANY 

(99) 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


AUG    4    1347 


LD  21-100m-12, '43  (8796s) 


4^0 


YB   17415 


M305S86 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


